Group theory

A group records reversible operations that can be combined consistently. Group theory studies their subgroups, group homomorphisms, group actions, and quotient groups. A symmetry group makes the abstraction visible: transformations may change individual points while preserving a specified object or its structure. An orbit of a group action records which positions can be exchanged, while a stabilizer subgroup records which operations leave a selected feature fixed.

H \leq G

and a normal subgroup

N ⊴ G

, the map

h \mapsto h N

from

H

onto

H N / N

is a group homomorphism with kernel

H \cap N

. The first isomorphism theorem gives the displayed group isomorphism. The product

H N

is a subgroup because

N

is normal. In particular, if

H N = G

and

H \cap N = 1

, then

H ≅ G / N

Transfer (group theory, $V_{G, H} : G ⟶ H / H^{'}$ )

 1  0

For a subgroup

H

of finite index, choose representatives

t

of right cosets

H t

. Write

t g = h (t, g) t_{g}

with

h (t, g) \in H

. The transfer is

V_{G, H} (g) = \prod_{t} h (t, g) (mod H^{'})

. Replacing

t

a_{t} t

multiplies each factor by

a_{t}

on the left and

a_{t_{g}}^{- 1}

on the right; these cancel in the abelianization because

t \mapsto t_{g}

is a permutation. The identity

h (t, g_{1} g_{2}) = h (t, g_{1}) h (t_{g_{1}}, g_{2})

proves the group homomorphism property after multiplying all factors.

Burnside transfer theorem ( $P \subseteq Z (N_{G} (P)) ⟹ G has a normal p -complement$ )

 0  0

The hypothesis makes

P

abelian and makes any two conjugate elements of

P

equal. For the latter assertion, Sylow conjugacy inside the centralizer of the target element adjusts a conjugating element into

N_{G} (P)

, where it centralizes

P

. Evaluating group transfer to

P

on an element

u \in P

, each coset cycle contributes a conjugate of

u

raised to the cycle length, hence that same power of

u

. The restriction is consequently

u \mapsto u^{[G : P]}

, an automorphism of the abelian

p

-group. Transfer is surjective, and its kernel is the required normal p-complement.

Cyclic Sylow subgroup at the least prime gives a normal complement

 0  0

Let

p

be the smallest prime dividing the order of a finite group, and suppose its Sylow subgroup

P

is cyclic of order

p^{a}

. Conjugation embeds

N_{G} (P) / C_{G} (P)

into

Aut (P)

, of order

p^{a - 1} (p - 1)

. Since

P \subseteq C_{G} (P)

, this quotient has order prime to

p

. Its remaining possible prime divisors are smaller than

p

and cannot divide

∣ G ∣

. Thus

N_{G} (P) = C_{G} (P)

, and the Burnside transfer theorem gives a normal p-complement.

Prüfer rank ( $rk_{P} (G) = sup_{H \leq G, H finitely generated} d (H)$ )

 1  0

The Prüfer rank bounds the number of generators of a group needed for each finitely generated subgroup, without requiring the ambient group to be finitely generated. The additive group

Q

has Prüfer rank one: any finitely generated subgroup has a common denominator and is cyclic, but no finite collection generates all rational numbers. Prüfer rank cannot increase under taking subgroups or quotients; for quotients, lift a finite generating set and apply the rank bound to the subgroup generated by its lifts.

Prüfer rank bound for extensions ( $rk_{P} (G) \leq rk_{P} (N) + rk_{P} (G / N)$ )

 0  0

For a normal subgroup

N

with Prüfer rank

r

and quotient of rank

s

, let

H = ⟨ h_{1}, \dots, h_{k} ⟩ \leq G

. Choose at most

s

elements

x_{j} \in H

generating the image of

H

. Write

h_{i} = w_{i} (x_{1}, \dots, x_{s}) n_{i}

with

n_{i} \in N \cap H

. The finitely generated subgroup

⟨ n_{1}, \dots, n_{k} ⟩

needs at most

r

generators, so

H

needs at most

r + s

. The full intersection

H \cap N

need not be finitely generated.

Maximal condition on subgroups ( $max$ )

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A group has the maximal condition on subgroups when every ascending sequence of subgroups eventually stabilizes. Equivalently, every subgroup is a finitely generated group. If a subgroup were not finitely generated, repeatedly adjoining an element outside the previously generated subgroup would give a strictly ascending sequence. Conversely, generators of the union of an ascending sequence all lie in one term, making that term the whole union. This condition is preserved by group extensions because each subgroup is an extension of its intersection with the kernel by its image in the quotient.

Group ( $G$ )

 1  0

A group packages reversible operations together with a rule for composing them. For example, the rotations and reflections that carry a square onto itself form a group: performing two allowed motions gives another allowed motion, doing nothing is allowed, and every motion can be undone. The elements are the motions themselves, rather than the vertices they move. Labels in the figure track those vertices; they are markers, rather than extra decorations that the motions must preserve.

Figure 1.
Square symmetries composed in opposite orders
. The rotation $r$ is counterclockwise through $9 0^{\circ}$ , and the reflection $s$ is in the vertical axis. Products act rightmost first: $rs = r \circ s$ . The red vertex starts at $(1, 1)$ and ends at $(- 1, - 1)$ under $rs$ , but at $(1, 1)$ under $sr$ . Both motions preserve the unmarked square, while their permutations of its vertices differ. **Details**.

Precisely, a group is a set

G

equipped with a binary operation

G \times G \to G

, written

(a, b) \mapsto ab

, satisfying the group axioms:

Closure: $ab \in G$ for all $a, b \in G$ . This is already part of specifying the operation's target as $G$ .
Associativity: $(ab) c = a (b c)$ for all $a, b, c \in G$ .
An identity element $e \in G$ satisfies $e a = a e = a$ for every $a \in G$ .
Every $a \in G$ has an inverse element $a^{- 1} \in G$ with $a a^{- 1} = a^{- 1} a = e$ .

The group operation need not be ordinary numerical multiplication. With addition as the operation, the identity element is usually written

0

and the inverse element of

a

is written

- a

. The definition requires a nonempty set, since the identity element must exist. See McKernan's MIT lecture 1, Definition 1.1 for the axioms.

Why insist on associativity? It makes a string of operations unambiguous without specifying parentheses. For function composition, both

(f g) h

and

f (g h)

send

x

f (g (h (x)))

. This fixes the order of execution while allowing regrouping. Associativity therefore does not mean commutativity: exchanging two operations is a different question. Ordinary subtraction fails associativity, since

(5 - 3) - 1 = 1

whereas

5 - (3 - 1) = 3

; it cannot serve as a group operation on the integers.

An inverse element makes reversal exact and equations solvable. In any group,

a x = b

has the unique solution

x = a^{- 1} b

, since

a^{- 1} (a x) = (a^{- 1} a) x = x

. Similarly,

x a = b

has solution

x = b a^{- 1}

. The side matters in a non-abelian group. Undoing a composite operation requires reversing its order:

(ab)^{- 1} = b^{- 1} a^{- 1}, (ab) (b^{- 1} a^{- 1}) = a (b b^{- 1}) a^{- 1} = e .

(1)

The analogous product in the other order also equals the identity element. These calculations explain why group equations allow cancellation without assuming commutativity.

For the square with vertices

(\pm 1, \pm 1)

, choose the rotation and reflection

r (x, y) = (- y, x), s (x, y) = (- x, y) .

(2)

Throughout this example, multiplication is function composition with the rightmost operation performed first:

(g h) (v) = g (h (v))

. Our notation

D_{4}

means the dihedral group of the square, with eight elements; the alternative convention calls the same group

D_{8}

. Its elements are

D_{4} = {e, r, r^{2}, r^{3}, s, rs, r^{2} s, r^{3} s}, r^{4} = e, s^{2} = e, srs = r^{- 1} .

(3)

Here

r^{- 1} = r^{3}

is clockwise rotation through

9 0^{\circ}

, while each reflection is its own inverse element. There are exactly eight motions: the image of one vertex has four choices, and the image of an adjacent vertex has two choices; together these determine the motion. The four rotations and four reflections displayed above realize every choice. The vertex permutations give a concrete group action. The general polygon construction is discussed in Group Theory (v4.01) by James S. Milne (section 1.17).

Computing the two products rather than merely comparing their names gives

(rs) (x, y) = r (- x, y) = (- y, - x), (sr) (x, y) = s (- y, x) = (y, x) .

(4)

Thus

(rs) (1, 1) = (- 1, - 1)

but

(sr) (1, 1) = (1, 1)

, proving

rs \neq = sr

. In fact

rs

is reflection in the line

y = - x

, whereas

sr

is reflection in

y = x

. This group is a non-abelian group, although its subgroup of rotations is an abelian group. The relation

srs = r^{- 1}

expresses a geometric fact: conjugation of a counterclockwise rotation by a reflection turns it clockwise.

A contrasting example is the cyclic group

Z /4 Z

under addition in modular arithmetic. Using representatives

0, 1, 2, 3

, the sum

3 + 2

1

modulo

4

, and the inverse element of

3

1

. Repeated addition of

1

runs through

0, 1, 2, 3, 0

. The correspondence

k \mapsto r^{k}

identifies this cyclic group with the rotation subgroup of

D_{4}

r^{3} r^{2} = r^{5} = r

. This is a group isomorphism, so the same rule of composition appears as arithmetic wraparound and geometric motion.

By comparison,

(Z, +)

is an infinite cyclic group. It has identity element

0

, inverse element

- k

for each integer

k

, and a generator of a group

1

, but no wraparound:

3 + 2 = 5

, and no positive number of copies of

1

sums to

0

. Both examples are abelian groups. The nonnegative integers under addition have closure, associativity, and an identity element, but lack an inverse element for

1

. Reversibility is precisely what fails.

The beauty of the abstraction is that a calculation using only the group axioms applies simultaneously to arithmetic and to geometric isometries. To make that connection universal, let each

g \in G

act on

G

by the translation in a group

L_{g} (x) = gx

. Each

L_{g}

is a bijection, with inverse

L_{g^{- 1}}

, and

L_{g} L_{h} = L_{g h}

. Distinct elements give distinct maps because

L_{g} (e) = g

. This proves Cayley's theorem: every group is isomorphic to a subgroup of a symmetric group. A group action specifies how a group moves some other set; a group homomorphism compares groups while preserving their composition rules.

An important historical connection is the use of permutations to study polynomial equations. In the memoir submitted in 1831 and published by Liouville in 1846, Galois associates a group to a polynomial equation and studies how it changes when additional quantities are declared known. Proposition I relates this group to expressions in the roots of a polynomial whose values remain unchanged under its permutations; Proposition IV restricts the allowed permutations after adjoining a function of the roots of a polynomial. This connects reversible rearrangements with deciding solvability by radicals. See Galois's original memoir, pp. 417–433, especially Propositions I and IV; Liouville's introduction, p. 382.

Partially ordered group

 1  0

A group with a partial order preserved by multiplication on both sides:

a \leq b

implies

x a y \leq x b y

for all

x, y

. In additive notation this says

a \leq b \Rightarrow x + a + y \leq x + b + y

. Preserving only one side is a different orderability condition.

Positive cone of an ordered group ( $G_{\geq 1} = {g : 1 \leq g}$ )

 0  0

The nonnegative cone determines the partial order by

a \leq b ⟺ a^{- 1} b \in G_{\geq 1}

. It contains the identity element, is closed under products and conjugation, and intersects its inverse only in the identity. Conversely these conditions define a two-sided invariant partial order. Its strictly positive part omits the identity; conventions for

G^{+}

differ, so specify which is intended.

Convex subgroup of an ordered group

 0  0

A subgroup

C

is convex if

c_{1} \leq g \leq c_{2}

with

c_{1}, c_{2} \in C

implies

g \in C

. In a linearly ordered group, convex subgroups form a chain under inclusion: positive elements witnessing incomparability would be comparable and force the smaller one into the other subgroup by convexity.

Lattice-ordered group ( $ℓ -group$ )

 0  0

An ordered group whose partial order is a lattice. Each pair has a join

f \lor g

and a meet

f \land g

. Translations preserve these operations, for example

a (f \lor g) b = a f b \lor a g b

. Pointwise maximum and minimum make the order automorphisms of a chain into a lattice-ordered group.

Riesz decomposition in a lattice-ordered group ( $1 \leq x \leq ab ⟹ x = a^{'} b^{'}, 1 \leq a^{'} \leq a, 1 \leq b^{'} \leq b$ )

 0  0

For

a, b \geq 1

, take

a^{'} = x \land a

and

b^{'} = (a^{'})^{- 1} x

. Then

a^{'} \geq 1

and

b^{'} = 1 \lor a^{- 1} x

is between one and

b

. Thus

x = a^{'} b^{'}

. Repetition splits a positive element bounded by a finite product into factors bounded by the corresponding factors. Every resulting positive factor is also bounded by

x

Lattice-ordered group homomorphism

 0  0

A group homomorphism preserving both join and meet. A bijective such map is an isomorphism of lattice-ordered groups; an injective one is an embedding.

Absolute value in a lattice-ordered group ( $∣ g ∣ = g \lor g^{- 1}$ )

 0  0

The displayed join is nonnegative in a lattice-ordered group. In an additive Abelian group it becomes

∣ g ∣ = g \lor (- g)

. This order-theoretic absolute value need not be a real scalar.

Positive and negative parts in a lattice-ordered group ( $g^{+} = g \lor 1, g^{-} = g^{- 1} \lor 1$ )

 0  0

The two positive parts satisfy

g = g^{+} (g^{-})^{- 1}

, commute with one another, and have meet one. They isolate positive and negative displacement in the pointwise model of a lattice-ordered permutation group. These are group elements, unlike the scalar positive part of a real number.

Convex lattice subgroup

 0  0

A convex subgroup of an ordered group that is also closed under join and meet. A normal convex lattice subgroup is the kernel of the quotient lattice-ordered group homomorphism.

One-sided bounds for adjoining a positive element ( $⟨ H, a ⟩_{c} = {g : ∣ g ∣ \leq h a^{n}} = {g : ∣ g ∣ \leq a^{n} h}$ )

 0  0

Assume

uv \leq v^{2} u^{2}

for all

u, v \geq 1

in a lattice-ordered group. For a convex lattice subgroup

H

and

a \geq 1

, the displayed bounds allow

h \in H

with

h \geq 1

and integers

n \geq 0

. To prove the first set is a subgroup, use

(h a^{n}) (k a^{m}) \leq h k^{2} a^{2 n + m}

. Products and inverse products are both bounded because

x y \leq ∣ x ∣∣ y ∣

and

(x y)^{- 1} \leq ∣ y ∣∣ x ∣

. Taking the larger exponent and a join of the two bounds in

H

bounds their absolute value in a lattice-ordered group. Inversion preserves that absolute value; common bounds using

h \lor k

prove lattice closure and convexity. The set contains

H

and

a

, and any convex lattice subgroup containing them contains the entire set. The second equality follows by applying the proof to the opposite group, where the required inequality is the original one with the variables interchanged.

Ordered right cosets of a convex lattice subgroup ( $P g \leq P h ⟺ g \leq p h for some p \in P$ )

 0  0

The displayed relation makes the right cosets into a lattice, with

P g \lor P h = P (g \lor h)

and

P g \land P h = P (g \land h)

. Antisymmetry uses convexity:

g \leq p h

and

h \leq q g

give

1 \leq p h g^{- 1} \leq pq \in P

, hence

h g^{- 1} \in P

. A prime convex lattice subgroup makes this order total even if the subgroup is not normal.

Lattice of convex lattice subgroups

 0  0

The convex lattice subgroups of a lattice-ordered group form a distributive lattice, with intersection as meet and the convex lattice subgroup generated by a union as join. A positive element of a join is bounded by a finite product of positive elements from the generating subgroups. Riesz decomposition in a lattice-ordered group splits such an element into positive factors from those subgroups. If the original element belongs to another convex subgroup

E

, every factor does too. Consequently

E \cap (C \lor D) = (E \cap C) \lor (E \cap D)

Principal convex lattice subgroup ( $⟨ u ⟩_{c} = {g : u^{- n} \leq g \leq u^{n} for some n}$ )

 0  0

For

u \geq 1

in a lattice-ordered group, the displayed set is the smallest convex lattice subgroup containing

u

. Products of two such bounds are bounded by a larger power of

u

; inversion and the lattice operations preserve the bounds. This gives an explicit test for membership.

Prime convex lattice subgroup

 0  0

A proper convex lattice subgroup

P

is prime when

a, b \geq 1

and

a \land b = 1

imply

a \in P

b \in P

. Equivalently the ordered right cosets of

P

form a totally ordered set. Prime refers to this order property, and does not require a normal subgroup or prime index.

Value in a lattice-ordered group

 0  0

A value of

g \neq = 1

is a convex lattice subgroup maximal among those omitting

g

. Zorn's lemma supplies a value: the union of a chain of convex lattice subgroups omitting

g

still omits

g

. Such subgroups are prime. Their unique cover is the smallest convex lattice subgroup properly containing them.

Squared bound from comparison at values ( $V pq \geq V x for all values of x ⟹ p^{2} q^{2} \geq x$ )

 0  0

Let

p, q, x \geq 1

in a lattice-ordered group. Under the displayed hypothesis, suppose

d = (x q^{- 2} p^{- 2}) \lor 1 > 1

and choose a value

P

d

. If

x \in P

, then

1 \leq d \leq x

contradicts its omission. Otherwise extend

P

to a value

V

x

. The right-coset test is

V u \leq V v ⟺ (u v^{- 1}) \lor 1 \in V

. If

p \in / V

, put

c = (x q^{- 2} p^{- 1}) \lor 1 \in V

. Then

(p c^{- 1}) \lor 1 \in / V

, while its inverse positive part is

(c p^{- 1}) \lor 1 = d

. These parts have meet one, so primeness of

P

forces

d \in P

. If

p \in V

, put

c = (x q^{- 1}) \lor 1 \in V

and

t = p^{2} q \in / V

; the disjoint pair

(t c^{- 1}) \lor 1

and

(c t^{- 1}) \lor 1 = d

gives the same contradiction. No normality assumption is used in this separation argument.

Cover of a value in a lattice-ordered group ( $V^{*} = ⟨ V, t ⟩_{c}$ )

 0  0

V

is a value in a lattice-ordered group of

t > 1

, every convex lattice subgroup strictly containing

V

contains

t

. Thus the convex lattice subgroup generated by

V

and

t

is the unique least subgroup properly containing

V

. It is called the cover of

V

. There is no intermediate convex lattice subgroup. A normal-valued lattice-ordered group requires

V

to be a normal subgroup of this cover, not necessarily of the whole group. Values are prime convex lattice subgroups: maximality omitting

t

makes them meet-irreducible in the lattice of convex lattice subgroups, so their ordered right cosets form a chain.

Power cofinality in the cover of a value ( $Vb \leq V a^{n}, bV \leq a^{m} V$ )

 0  0

Under the inequality in one-sided bounds for adjoining a positive element, if

a \geq 1

belongs to the cover of a value in a lattice-ordered group

V^{*}

but not to

V

, then

⟨ V, a ⟩_{c} = V^{*}

. Thus every

b \geq 1

in the cover is bounded above by both

v a^{n}

and

a^{m} w

for suitable nonnegative

v, w \in V

and integers

m, n \geq 0

. The corresponding right and left coset bounds give the displayed cofinality, without first assuming that

V

is normal. Left-coset order is obtained by inversion of right-coset order.

Normal-valued lattice-ordered group

 0  0

A lattice-ordered group is normal-valued when every value in a lattice-ordered group is a normal subgroup of its cover. Normality in the cover is weaker than normality in the entire group. Every linearly ordered group is normal-valued.

Wolfenstein inequality for normal-valued lattice-ordered groups ( $∣ f ∣∣ g ∣ \leq ∣ g ∣^{2} ∣ f ∣^{2}$ )

 0  0

A lattice-ordered group is normal-valued precisely when the displayed inequality holds for all its elements. The theorem is proved without using it as a definition. In the forward direction a normal value has a linearly ordered quotient of its cover with no proper nontrivial convex subgroup. That quotient is Archimedean, hence Abelian by Archimedean linearly ordered groups are Abelian. Apply squared bound from comparison at values to

p = b, q = a, x = ab

. For the reverse direction, power cofinality in the cover of a value excludes a conjugate

u = a^{- 1} v a

outside the value: its right cosets

V u^{n}

are all bounded by

Va

, contradicting cofinality above

V a^{2}

. Left-coset cofinality treats

a v a^{- 1}

. Every element is a quotient of nonnegative lattice parts, proving normality in the cover. In a linearly ordered group, comparison of

a, b \geq 1

immediately gives

ab \leq max (a, b)^{2} \leq b^{2} a^{2}

, so every such group is normal-valued.

Linearly ordered group

 1  0

A group with a two-sided invariant total order. Thus it is a lattice-ordered group whose join and meet choose the larger and smaller element. An order on only one side is not sufficient here. Every linearly ordered group is torsion-free: a positive element has strictly increasing positive powers, and a negative element has a positive inverse.

Archimedean ordered group

 0  0

An ordered group is Archimedean when

a \geq 1

and

a^{n} \leq b

for all positive integers force

a = 1

. In an additive Abelian group this excludes a positive element whose every integer multiple stays below one fixed bound. This generalizes the Archimedean property of the real numbers.

Archimedean linearly ordered groups are Abelian

 0  0

An Archimedean ordered group with a two-sided invariant total order is commutative. If there is a least positive element, its cofinal powers exhaust the group. Otherwise each

c > 1

has

d > 1

with

d^{2} < c

. To see this, choose

1 < s < r < c

and use

r s^{- 1}

if its square is below

r

; if not, the inequality

(r s^{- 1})^{2} \geq r

gives

s^{2} \leq r < c

. If positive

g, h

satisfied

g h < h g

, put

c = (g h)^{- 1} h g

and choose such a

d

. Cofinality gives

d^{m} \leq g < d^{m + 1}

and

d^{n} \leq h < d^{n + 1}

. Two-sided order invariance then gives both

h g < d^{m + n + 2}

and

h g = g h c > d^{m + n + 2}

, a contradiction. Replacing negative elements by their inverses reduces arbitrary commutativity to the positive case.

Lexicographically ordered group

 0  0

For ordered Abelian groups, compare tuples at their first differing coordinate. This gives a two-sided invariant total order on their direct product. The additive group

Z^{2}

with its first coordinate dominant is not an Archimedean ordered group, since

n (0, 1) < (1, 0)

for every positive integer

n

Convex subgroups of a finite-rank ordered Abelian group ( $# {convex subgroups} \leq r + 1$ )

 0  0

For a torsion-free abelian group of finite rational rank

r

, each convex subgroup of an ordered group is pure: if

n g \in C

for a positive integer

n

, then

g

lies between zero and

n g

or their negatives and hence belongs to

C

. A proper inclusion of convex subgroups therefore increases rational rank. They form a chain, so there are at most

r + 1

, including zero and the whole group. A lexicographically ordered copy of

Z^{r}

attains the bound.

Ohnishi orderability criterion

 0  0

A group admits a two-sided invariant total order exactly when every finite list of nonidentity elements can be signed so that the conjugation-invariant semigroup generated by the signed elements excludes the identity. A compactness theorem argument supplies a compatible global sign assignment. Omitting the conjugates changes the criterion to a different orderability problem.

Abelian lattice-ordered group

 0  0

A commutative lattice-ordered group, usually written additively. Examples include real-valued function groups with pointwise addition, maximum and minimum. Such a group need not have a total order.

Hahn group ( $V (Γ, R)$ )

 0  0

For a totally ordered set

Γ

, take real coefficient families whose support satisfies the maximum condition: every nonempty subset of the support has a largest index. Add coefficients pointwise and decide the sign from the largest nonzero coordinate. This defines an ordered Abelian group. For

Γ = Z_{> 0}

with its usual order, the maximum condition forces finite support. Some presentations reverse the index order and use a minimum instead; specify the convention when applying an embedding theorem.

Finite-support Hahn embedding along a countable convex chain ( $G ≅ ⨁_{n \geq 1} R$ )

 0  0

Let a divisible ordered Abelian group be the union of a strictly increasing sequence of convex subgroups

C_{1} = 0, C_{2}, \dots

, with each ordered quotient

C_{n + 1} / C_{n}

isomorphic to the additive real line. Convexity makes every

C_{n}

divisible, so these are rational vector subspaces. Choose a rational-linear section for each quotient. Every element then has a unique finite sum of section values. Its sign is the sign of the coefficient of the largest active index, because convexity makes the corresponding ordered quotient decide its sign. The coefficients give an embedding, indeed an ordered-group isomorphism, into the Hahn group with positive integer indices in the maximum-support convention. No compatibility of a separate real scalar multiplication is needed for this group embedding.

Free Abelian lattice-ordered group

 0  0

The free object on generators

x_{1}, \dots, x_{n}

has the universal property that an arbitrary assignment of these generators to an Abelian lattice-ordered group extends uniquely to a lattice-ordered group homomorphism. It can be constructed from terms using addition, negation, zero, join and meet, modulo identities valid in every Abelian lattice-ordered group. Its underlying group need not be finitely generated even when there are only two lattice generators.

Piecewise integer-linear representation of a free Abelian lattice-ordered group ( $F_{n} = ⟨ x_{1}, \dots, x_{n} ⟩_{ℓ} \subseteq R^{R^{n}}$ )

 0  0

Evaluate the generators as coordinate projections on

R^{n}

. The lattice-ordered group they generate consists of continuous positively homogeneous piecewise linear functions with finitely many integer-linear pieces. Equivalently its elements are finite joins of finite meets of integer linear forms. The universal property follows because the real line detects every identity of Abelian lattice-ordered groups: if an identity fails, pass to a prime ordered quotient and select the finitely many active linear pieces. The resulting rational linear inequalities have a real solution by the alternative underlying Farkas' lemma, giving a real witness to failure. This is a statement about integer coefficients and positive homogeneity, not arbitrary real coefficients or affine constant terms.

Integer-linear hinge functions have infinite independent rank ( $h_{n} (x, y) = max (0, x - n y)$ )

 0  0

The piecewise linear functions

h_{n}

, for distinct integer indices, are linearly independent over the integers. Restrict a finite integer relation to

y = 1

. At

x = n

, only

h_{n}

has a jump of one in its slope; hence the coefficient of that function must be zero. This proves that the underlying group of the two-generated free Abelian lattice-ordered group has infinite rank of an abelian group.

Lattice-ordered permutation group

 0  0

A group of order automorphisms of a totally ordered set that is closed under pointwise maximum and minimum. Products act on the right when writing

α f

; then

α (f g) = (α f) g

. The pointwise lattice operations make it a lattice-ordered group.

Order-primitive lattice permutation group

 0  0

A transitive lattice-ordered permutation group is order-primitive when its only invariant equivalence relations with convex classes are equality and the universal relation. This concerns convex blocks in a chain, rather than arbitrary blocks of a primitive group action.

McCleary trichotomy theorem

 0  0

A faithful transitive order-primitive lattice permutation group is either a regular Archimedean translation action, an order two-transitive action, or a periodic action. In the periodic case a fixed-point-free order automorphism of the Dedekind completion commutes with the action and has cofinal and coinitial iterates; the point stabilizers act order two-transitively within period intervals. Only the regular case satisfies

∣ f ∣∣ g ∣ \leq ∣ g ∣^{2} ∣ f ∣^{2}

for every pair. In either other case, order two-transitivity on the whole chain or on a period interval supplies positive

a, b

and

x < y < z < t

with

x a = y

y a = z

x b = x

y b = t

, giving

x ab = t > z = x b^{2} a^{2}

. The classification itself is a theorem, not a consequence of that witness calculation.

Holland representation theorem

 0  0

Every lattice-ordered group embeds in a lattice-ordered permutation group on one totally ordered set. Choose a value in a lattice-ordered group for each nonidentity element, act on its ordered right cosets, and combine these chains in an order sum. The original element moves its own value's identity coset, making the action faithful. The coset lattice formulas make the embedding preserve both join and meet.

Order n-transitivity

 0  0

An ordered permutation group is order n-transitive if it can send any increasing

n

-tuple to any other increasing

n

-tuple. This compares tuples with the same relative order, unlike unrestricted n-transitive group actions. In the usual transitive chain setting, order two-transitivity implies every finite order-transitivity for a lattice-ordered permutation group.

Finite interpolation by lattice operations ( $h = ⋁_{i} ⋀_{j} h_{ij}$ )

 0  0

Suppose

h_{ij}

sends the

i

th and

j

th nodes of an increasing input tuple to the corresponding output nodes. At the

i

th input, every

h_{ij}

agrees, so their pointwise minimum takes the required value. At every other input it is bounded above by the required value, using the corresponding

j

term. The pointwise maximum over

i

therefore interpolates the entire tuple. The construction uses closure under finite join and meet, without assuming arbitrary restrictions of group elements belong to the group.

Positive bump of an order automorphism

 0  0

A positive bump of the real line is an order automorphism whose nonempty support of a permutation is one open interval and which moves every point of that interval to the right. It fixes the complement. Iterating any interior point tends to the right endpoint in positive time and to the left endpoint in negative time.

Conjugacy of positive real-line bumps

 0  0

Choose fundamental intervals

[x, x f)

and

[y, y g)

for two positive bumps of an order automorphism. Any increasing bijection matching their endpoints extends by

b (x f^{n}) = (x b) g^{n}

to their support intervals. Extend it monotonically over the two complementary rays. Then

f b = b g

, so

b^{- 1} f b = g

. Bounded supports impose no further invariant for positive bumps in the full real-line order-automorphism group.

Conjugating positive bumps while preserving a fundamental interval

 0  0

If two positive bumps of an order automorphism move a point

c

beyond an interval

[c, d]

, their conjugacy can be chosen to fix that interval: begin with the identity there, complete the increasing map between fundamental intervals, and extend equivariantly along bump iterates. If both bump supports lie in one fundamental interval of another positive motion

h

, fix that cell's endpoints and repeat the conjugator on all its

h

-translates. The result commutes with

h

, and with any automorphism supported in the interval fixed pointwise.

Group structure on every nonempty set ( $x ⋆ y = j^{- 1} (j (x) j (y))$ )

 0  0

Assuming choice, every nonempty set admits a group operation. A finite set receives a cyclic group structure. For an infinite set, a maximal partial group obtained by Zorn has infinite carrier

A

with smaller complement; infinite-cardinal doubling implies the whole set is equipotent with

A

. Transport its operation along a bijection

j

. This final group need not extend a previously chosen partial group on its original subset; transport changes labels and only asserts existence.

Exact factorization of a group ( $G = H_{1} H_{2}, H_{1} \cap H_{2} = {e}$ )

 0  0

Two subgroups give an exact factorization of a group when every element has a unique expression

h_{1} h_{2}

with

h_{i} \in H_{i}

. Existence and trivial intersection are equivalent to this uniqueness: if

h_{1} h_{2} = k_{1} k_{2}

, then

k_{1}^{- 1} h_{1} = k_{2} h_{2}^{- 1} \in H_{1} \cap H_{2}

, so both sides are the identity. Conversely, an element

a

in the intersection gives the two factorizations

a \cdot e = e \cdot a

, forcing

a = e

. For finite groups the multiplication map

H_{1} \times H_{2} \to G

is therefore a bijection and

∣ G ∣ = ∣ H_{1} ∣∣ H_{2} ∣

. It need not be a group homomorphism: an exact factorization is not automatically a direct product of groups. If

H_{1}

is normal, it gives an internal semidirect product.

Shifted addition on odd integers ( $a * b = a + b + 1$ )

 0  0

The odd integers form an abelian group under the displayed operation. The identity element is

- 1

and the inverse element of

a

- a - 2

. The bijection

a \mapsto (a + 1) /2

is a group isomorphism to

(Z, +)

, since it converts the shifted operation into addition. This illustrates how an unfamiliar identity and inverse can arise from ordinary addition by a change of coordinates.

Triangle group ( $Δ (p, q, r)$ )

 1  0

A triangle group is generated by reflections in the sides of a triangular fundamental domain in spherical geometry, Euclidean geometry or hyperbolic geometry. The angles

π / p, π / q, π / r

determine the geometry through the sign of

1/ p + 1/ q + 1/ r - 1

. Its orientation-preserving subgroup has group presentation

⟨ x, y, z ∣ x^{p} = y^{q} = z^{r} = x yz = 1 ⟩

Poly-(cyclic or finite) group

 0  0

A group is poly-(cyclic or finite) when it has a finite subnormal series

1 = G_{0} ◃ G_{1} ◃ \dots ◃ G_{r} = G

in which each quotient

G_{i} / G_{i - 1}

is a cyclic group or a finite group. Each term need only be normal in the next term. The polycyclic group condition instead requires every factor to be cyclic.

Group element

 0  0

A group element is a member of the underlying set of a group. The group operation assigns a product to each pair of elements, the identity fixes every element under multiplication, and every element has an inverse.

Integer power of a group element ( $x^{n} (n \in Z)$ )

 0  0

For an element

x

of a group with identity element

e

, define

x^{0} = e

x^{n}

as a product of

n

copies of

x

for positive

n

, and

x^{- n} = (x^{- 1})^{n}

. Associativity and cancellation of inverse elements give

x^{m + n} = x^{m} x^{n}

and

(x^{m})^{n} = x^{mn}

for all integers

m, n

. Consequently

n \mapsto x^{n}

is a group homomorphism from the additive group of integers onto the cyclic subgroup generated by

x

Braid group ( $B_{n}$ )

 1  0

The group of braids on

n

strands, composed by stacking, with adjacent-crossing generators

σ_{i}

and the braid group relations. It retains over/under crossings and is not a permutation group.

Artin automorphism of a braid ( $x_{i} \mapsto x_{i} x_{i + 1} x_{i}^{- 1}, x_{i + 1} \mapsto x_{i}$ )

 0  0

A positive elementary braid acts on the free group of punctured-disk meridians by the displayed substitution, fixing other generators. The inverse sends

x_{i}

x_{i + 1}

and

x_{i + 1}

x_{i + 1}^{- 1} x_{i} x_{i + 1}

. Applying Seifert-van Kampen theorem at the crossings and closing the braid gives the knot-complement presentation

⟨ x_{i} ∣ x_{i} = ϕ (x_{i})⟩

Braid group relations ( $σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}$ )

 0  0

The generators

σ_{i}

commute when

∣ i - j ∣ > 1

and satisfy

σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}

. Unlike a symmetric-group presentation, there is no

σ_{i}^{2} = 1

relation.

Infinite group

 1  0

An infinite group is a group with infinitely many elements. An element of infinite order proves that its group is infinite, but the converse need not hold: an infinite group can have all elements of finite order.

Tarski monster group

 1  0

An infinite group whose every proper nontrivial subgroup is cyclic of a fixed prime order

p

. Such groups exist for sufficiently large primes. This subgroup condition is stronger than the condition that every nonidentity element has order

p

// Target: algebra.bigb

Tarski monster groups are two-generated and simple

 0  0

For a Tarski monster group, any nonidentity element generates a proper subgroup of order

p

. Adjoining an element outside that subgroup generates the whole group. A proper nontrivial normal subgroup would have order

p

; conjugation into its finite automorphism group forces it to be central. Combining it with another order-

p

cyclic subgroup would then give a proper subgroup of order

p^{2}

, a contradiction.

// Target: algebra.bigb

Trivial group ( ${1}$ )

 0  0

The trivial group has exactly one element, its identity. Every group homomorphism to it is trivial, and a group is trivial precisely when all elements in a generating set equal the identity.

Group of exponent two is abelian ( $g^{2} = 1 \forall g ⟹ ab = ba$ )

 0  0

If every element of a group squares to the identity, every element equals its inverse. Then

ab = (ab)^{- 1} = b^{- 1} a^{- 1} = ba

, so the group is an abelian group. The argument does not require a finite group.

Non-abelian group

 0  0

A non-abelian group is a group containing elements

a, b

for which

ab \neq = ba

Torsion-free group

 0  0

A torsion-free group is a group in which the identity element is the only element of finite order of a group element.

Countable torsion-free embedding with two conjugacy classes

 0  0

Every nontrivial countable torsion-free group embeds in a countable torsion-free group with precisely two conjugacy classes, the identity and all nonidentity elements. At each stage, enumerate all ordered pairs of nonidentity elements of the current group and successively adjoin HNN extensions identifying their infinite cyclic subgroups. Britton's lemma preserves embeddings and torsion in an HNN extension preserves torsion-freeness. Repeat on the resulting countable union, so pairs involving newly introduced elements are eventually included. A final increasing union is countable and torsion-free, and any two nonidentity elements become conjugate at a later stage. No effective enumeration of nonidentity elements is claimed.

Divisible group

 1  0

A divisible group is an abelian group

G

in which, for every positive integer

n

and every

g \in G

, some

h \in G

satisfies

nh = g

Divisibility by a prime ( $p A = A$ )

 0  0

An abelian group

A

p

-divisible when multiplication by

p

is surjective, equivalently

A / p A = 0

. This does not mean it has no

p

-torsion:

Q / Z

is divisible and has elements of every finite order. The Kummer cohomology divisibility criterion relates this property to Galois cohomology.

Torsion-free divisible Abelian group

 0  0

A divisible Abelian group is torsion-free precisely when division by each nonzero integer is unique. Defining rational scalar multiplication by

n ((m / n) a) = ma

identifies these groups with vector spaces over the rational numbers, including the zero space.

Group axioms

 0  0

The group axioms require closure and associativity of the binary operation, an identity element, and an inverse for every element.

One-sided group axioms criterion

 0  0

An associative binary operation

*

on a nonempty set

X

makes

X

a group if there is

e \in X

with

x * e = x

for every

x

, and every

x

has some

z

with

x * z = e

. No left identity element or left inverse element assumption is needed. First, an idempotent element of a semigroup

y

satisfies

y = e

: choose

z

with

y * z = e

and compute

y = y * e = y * (y * z) = (y * y) * z = y * z = e

. If

x * z = e

, then

(z * x) * (z * x) = (z * (x * z)) * x = (z * e) * x = z * x

, so

z * x = e

by the previous observation. Finally,

e * x = (x * z) * x = x * (z * x) = x * e = x

. Thus all group axioms follow.

Group axioms from two-sided local inverse identities

 0  0

Suppose a nonempty semigroup has, for each

a

, elements

l_{a}, r_{a}

satisfying

b l_{a} a = b

and

a r_{a} b = b

for every

b

. Then it is a group. For any fixed

a

, the element

e_{R} = l_{a} a

is a right identity element, and

e_{L} = a r_{a}

is a left identity element. Since

e_{L} e_{R} = e_{L} = e_{R}

, they coincide in a two-sided identity

e

. For each

a

, putting

b = e

in the two assumed identities gives

l_{a} a = e

and

a r_{a} = e

. Associativity then yields

l_{a} = l_{a} e = l_{a} (a r_{a}) = (l_{a} a) r_{a} = e r_{a} = r_{a}

, a two-sided inverse element. Thus the usual group axioms follow without an assumed identity or cancellation law.

Finite group

 1  0

A finite group is a group whose underlying set has finitely many elements. Its order of a finite group is the number of elements, counting the identity element. The group operation still has to satisfy all the group axioms: merely having a finite list of states does not make a group.

For example, the six congruence classes modulo six form a finite cyclic group under addition, whereas the permutations of three objects form the symmetric group

S_{3}

, also with six elements. The first is an abelian group; the second is a non-abelian group. Thus the order of a finite group is a useful first measurement, but the composition rule contains information that the count alone loses.

Finiteness makes group structure accessible to exact calculation. A Cayley table for a finite group of order

n

has

n^{2}

entries, and every group element eventually returns to the identity element when repeated. Lagrange's theorem strengthens this observation: the order of a group element, being the size of its cyclic subgroup, divides

n

. The cyclic group of order four and Klein four-group give a small example of how that constraint can distinguish different composition rules.

p-prime core of a finite group ( $O_{p^{'}} (G)$ )

 0  0

The p-prime core is the largest normal subgroup whose order is coprime to

p

. Products of normal subgroups of order coprime to

p

again have order coprime to

p

, so it exists and is characteristic.

p-core of a finite group ( $O_{p} (G)$ )

 0  0

The p-core is the largest normal p-subgroup of a finite group. The product of two normal p-subgroups is again a normal p-subgroup, so this subgroup exists and is characteristic.

p-constrained group

 0  0

Put

\overline{G} = G / O_{p^{'}} (G)

using the p-prime core of a finite group. A finite group is p-constrained if

C_{\overline{G}} (O_{p} (\overline{G})) \leq O_{p} (\overline{G})

. If

O_{p^{'}} (G) = 1

, the p-core of a finite group has its centralizer contained in itself. Then block idempotents centralize a normal p-subgroup places every block idempotent in

k O_{p} (G)

, a local ring, so

k G

has only the principal block.

Pi-group ( $π -group$ )

 0  0

For a set

π

of primes, a pi-group is a finite group whose order has no prime divisors outside

π

. The trivial group qualifies for every

π

Pi-element ( $π -element$ )

 0  0

A pi-element is an element whose order has no prime divisors outside a specified prime set

π

. The identity is a pi-element for every prime set. Conjugation preserves the property because it preserves element order.

Elementary group

 0  0

In finite-group character theory, an elementary group is a group that is p-elementary group for some prime

p

. Such groups are solvable and provide the subgroup tests in Brauer's characterization of characters. This does not mean elementary abelian group.

Prime-set decomposition of elementary groups ( $E = E_{π} \times E_{π^{'}}$ )

 0  0

Write an elementary group as

P \times C

with

P

a p-group and

C

cyclic of order prime to

p

. Split

C

into its cyclic pi-part and pi-complementary part. Attach

P

to whichever part contains

p

, producing a direct product of groups of a pi-group and a complementary-prime group.

p-elementary group ( $E = P \times C, P a p -group, C cyclic, p ∤ ∣ C ∣$ )

 0  0

A p-elementary group is a direct product of groups of a p-group and a cyclic group whose order is coprime to

p

. Either factor may be trivial. The coprime cyclic factor is central, a stronger requirement than merely having a cyclic subgroup with p-group quotient.

Ambivalent group

 0  0

A finite group is ambivalent when every element is conjugate to its inverse. Since every complex character satisfies

χ (g^{- 1}) = \overline{χ (g)}

, an ambivalent group has real values for every irreducible character. Conversely, if every irreducible value is real, irreducible characters separate conjugacy classes and force every

g

to be conjugate to

g^{- 1}

Monomial group

 0  0

A finite group is monomial if every irreducible character is a monomial character. Finite metabelian groups have this property.

Abelian intersection of irreducible inducing subgroups ( $K = ⋂ {H : Ind_{H}^{G} φ is irreducible for some φ}$ )

 0  0

In a finite monomial group, the displayed intersection is a normal subgroup. Every irreducible character is induced from a linear character of a subgroup containing

K

. On restriction to

K

, all conjugate inducing lines remain invariant, so

K^{'}

lies in every irreducible character kernel. The intersection of those kernels is trivial, proving that

K

is Abelian.

H

satisfying

H \cap H^{g} = {1}

for every

g \in / H

, with

H^{g} = g^{- 1} H g

. Such

H

is a Frobenius complement. The Frobenius kernel theorem turns the identity together with elements outside all conjugates of

H

into a normal subgroup

N

, giving

G = N ⋊ H

Frobenius kernel ( $N = (G ∖ ⋃_{g} H^{g}) \cup {1}$ )

 0  0

The Frobenius kernel consists of the identity and the elements lying in no conjugate of a Frobenius complement. The Frobenius kernel theorem proves that this conjugacy-invariant set is a normal subgroup, rather than assuming closure as part of its definition.

Frobenius kernel criterion by irreducible induction ( $Ind_{N}^{G} φ \in Irr (G) (1_{N} \neq = φ \in Irr (N))$ )

 0  0

For

1 < N ◃ G

proper, the displayed condition is equivalent to

N

being a Frobenius kernel. For a normal subgroup the induction norm is

∣ I_{G} (φ) : N ∣

. Trivial inertia for each nonprincipal character and Brauer's permutation lemma imply that every element outside

N

has trivial centralizer in

N

. This forces

N

to be a normal Hall subgroup; a complement is then fixed-point-free on

N

Commutator bijection from a fixed-point-free automorphism ( $y ⟼ α (y)^{- 1} y$ )

 0  0

Let

N

be a finite group and

α

an automorphism whose only fixed element is the identity. If

α (a)^{- 1} a = α (b)^{- 1} b

, then

α (b a^{- 1}) = b a^{- 1}

, so

a = b

. The displayed map is therefore a bijection. For conjugation by a nonidentity element of a Frobenius complement, it gives the commutator map

y \mapsto h^{- 1} y^{- 1} h y

Frobenius kernel theorem ( $N ◃ G, G = N ⋊ H$ )

 0  0

For a Frobenius complement

H

, a class function

θ

with

θ (1) = 0

satisfies

(θ^{G})_{H} = θ

. For each nontrivial

η \in Irr (H)

of degree

d

, the virtual character

(η - d 1_{H})^{G} + d 1_{G}

has norm one and positive degree by Frobenius reciprocity. It is therefore irreducible, restricts to

η

and is identically

d

on the candidate kernel

N

. The intersection of the kernels of these characters is exactly

N

: outside

N

, separation follows from the character of the regular representation of

H

. This proves normality;

N \cap H = 1

and

∣ N ∣ = [G : H]

then give the semidirect product.

Frobenius complement ( $H \cap H^{g} = {1} (g \in / H)$ )

 0  0

A Frobenius complement is the nontrivial proper subgroup

H

in a Frobenius group with the displayed trivial-intersection property. In the action on cosets of

H

, it is a point stabilizer; no nonidentity element fixes two distinct points.

Burnside's theorem ( $∣ G ∣ = p^{a} q^{b}$ )

 0  0

Every finite group whose order is divisible by at most two distinct primes is a solvable group. A nonabelian simple group must therefore have at least three distinct prime divisors.

Prime-power conjugacy-class obstruction to simplicity

 0  0

A nonabelian simple group has no nonidentity conjugacy class of prime-power size. Coprime-degree irreducible characters vanish on such an element: the conjugacy-class sum makes the character-to-degree ratio an algebraic integer, and the Kronecker theorem on algebraic integers in the unit disk makes a nonzero ratio a root of unity, forcing a scalar in a faithful group representation. Column character orthogonality then makes

1/ p

an algebraic integer, impossible. This is the character-theoretic ingredient in Burnside's theorem.

Abelian subgroup cannot have prime-power index in a nonabelian simple group

 0  0

If an abelian subgroup

A

has index

p^{a}

, the centralizer of any

1 \neq = g \in A

contains

A

, so the class size of

g

divides

p^{a}

. The prime-power conjugacy-class obstruction to simplicity rules this out. If

A = 1

, the group is a p-group and has nontrivial center of a group, also ruling out nonabelian simplicity.

Order of a finite group ( $∣ G ∣$ )

 0  0

The order of a finite group is its cardinality, written

∣ G ∣

. The identity element is included:

∣ Z /6 Z ∣ = 6

, and the trivial group has order one, rather than zero.

Keep this count distinct from the order of a group element. In the additive finite cyclic group

Z /6 Z

, the element

1

has order six,

2

has order three,

3

has order two, and

0

has order one. Each element's order is the least positive number of repetitions needed to return to the identity element

0

. The order of an element of a finite cyclic group computes these periods directly.

The count also constrains subgroups: Lagrange's theorem gives

∣ G ∣ = [G : H] ∣ H ∣

for a subgroup

H

of a finite group

G

, where the index of a subgroup counts its cosets. For

H = {0, 2, 4}

inside

Z /6 Z

, the two cosets are

{0, 2, 4}

and

{1, 3, 5}

, so

6 = 2 \cdot 3

. Divisibility is a necessary constraint; it does not imply that every divisor occurs as the order of a group element, as the Klein four-group shows.

For a nonempty inclusion chain of group operations on subsets of a fixed set, the union is a group operation on the union of their carriers. Any finite list of carrier elements lies in one chain member, so its multiplication and associativity agree there. Nested identities coincide because the only idempotent of a group is its identity; inverses remain valid in larger members. Thus the union supplies the chain upper bound required by Zorn's lemma.

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Group theory is a branch of mathematics that studies algebraic structures known as groups. A group is defined as a set equipped with a single binary operation that satisfies four fundamental properties: 1. **Closure**: If $ a $ and $ b $ are elements of the group, then the result of the operation $ a * b $ is also in the group.

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