Group theory

Group theory studies groups, homomorphisms, actions, quotients, and symmetry.

A group is a set with an associative binary operation, an identity, and an inverse for every element.

The group of braids on $n$ strands, composed by stacking, with adjacent-crossing generators ${\sigma }_i$ and the braid group relations. It retains over/under crossings and is not a permutation group.

The generators ${\sigma }_i$ commute when $|i-j|>1$ and satisfy ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$. Unlike a symmetric-group presentation, there is no ${\sigma }_i^2=1$ relation.

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.

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.

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.

A non-abelian group is a group containing elements $a,b$ for which $ab\ne ba$.

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

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 $nh=g$.

An abelian group $A$ is $p$-divisible when multiplication by $p$ is surjective, equivalently $A/pA=0$. This does not mean it has no $p$-torsion: $\mathbb{Q}/\mathbb{Z}$ is divisible and has elements of every finite order. The Kummer cohomology divisibility criterion relates this property to Galois cohomology.

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.

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

A finite group is a group with finitely many elements. Its number of elements is its order.

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.

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.

If an abelian subgroup $A$ has index $p^a$, the centralizer of any $1\ne g\in A$ 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.

The order $|G|$ of a finite group is its number of elements.

A nontrivial group has more than one element.

An identity element $e$ satisfies $eg=ge=g$ for every element $g$ of a group.

The inverse $g^{-1}$ of a group element $g$ satisfies $g{g}^{-1}=g^{-1}g=e$.

A group operation is the associative binary operation of a group. It has an identity element, and every element has an inverse element.

An operation is associative when $(xy)z=x(yz)$ whenever the products are defined.

The group commutator measures the failure of two elements to commute. Under the convention $[x,y]=x^{-1}{y}^{-1}xy$, it is the identity exactly when $xy=yx$.

The Hall-Witt identity is a three-variable identity among iterated group commutators. Passing to the leading terms of a filtered group turns it into the Jacobi identity for the associated graded Lie algebra.

The Hall-Petrescu formula expands $(xy{)}^{p^n}$ as $x^{p^n}{y}^{p^n}$ times ordered powers of higher commutators. In a p-valued group, the correction terms have controlled higher valuation and make the p-power map linear on the associated graded group.

Left and right multiplication by a fixed group element are bijections, called left and right translations.

A cyclic group is generated by one element. A cyclic group of order $n$ has exactly $d$ solutions of $x^d=1$ for every positive integer divisor $d$ of $n$.

The cyclic subgroup generated by an element $g$ is $⟨g⟩=\{g^n:n\in \mathbb{Z}\}$. Its size is the order of $g$.

In a finite group, a cyclic subgroup of order $m$ has exactly $\phi (m)$ generators, where $\phi$ is the Euler totient function. Each element generates exactly one cyclic subgroup, so dividing the number of elements of order $m$ by $\phi (m)$ counts these subgroups. Include the identity subgroup using $m=1$ and $\phi (1)=1$. In a symmetric group, element counts come from cycle type and order of a group element is the least common multiple of cycle lengths.

A finite cyclic group of order $n$ is isomorphic to $C_n=\mathbb{Z}/n\mathbb{Z}$.

For any group $G$, evaluation at a chosen generator of a group $x$ of $C_n$ gives a bijection between group homomorphisms $C_n\to G$ and elements satisfying $g^n=1$. A group homomorphism must send $x^j$ to $g^j$. Conversely, this formula is well-defined because exponents differing by a multiple of $n$ give equal powers, and it respects multiplication. The displayed correspondence is a bijection of sets; it is not claimed to be a group isomorphism for a nonabelian target. In a symmetric group, the order of a group element is the least common multiple of its disjoint cycle lengths, making this correspondence easy to enumerate.

An infinite cyclic group is generated by an element of infinite order and is isomorphic to the additive group of integers.

Every finite subgroup of ${\mathbb{C}}^{\times }$ is cyclic. If its exponent is $n$, it is a subgroup of the cyclic group of $n$th roots of unity.

A generator of a cyclic group $G$ is an element $g$ such that every element of $G$ is an integer power of $g$.

A subset $S$ of a group $G$ is a generating set when every element of $G$ is a finite product of elements of $S$ and their inverses, equivalently when the subgroup $⟨S⟩$ generated by $S$ equals $G$.

For a proper subgroup $H$ of any group $G$, its complement is a generating set of a group. Choose $x\notin H$. For every $h\in H$, also $xh\notin H$, and $h=x^{-1}(xh)$ lies in the subgroup generated by the complement. That subgroup consequently contains both $H$ and its complement. No finiteness assumption is needed.

A group is finitely generated when it has a finite generating set of a group.

If $G$ has $d$ generators, there are at most $(n!{)}^d$ group homomorphisms $G\to S_n$. Each subgroup of index $n$ is a point stabilizer in a transitive coset group action, so there are at most $n(n!{)}^d$ such subgroups. Consequently finitely many subgroups have index at most any fixed bound. Intersecting them gives a finite-index characteristic subgroup, useful in proving residual finiteness of semidirect products.

The rank $d(G)$ of a finitely generated group is the minimum cardinality of a generating set of a group of $G$.

An abelian group is a group whose operation is commutative: $gh=hg$ for every $g,h$ in the group.

A class of abelian groups closed under isomorphisms, subgroups, quotients and extensions. For the homotopical Serre-class theorems one also uses tensor/Tor closure and homology closure for Eilenberg–MacLane spaces. A homomorphism is an isomorphism modulo the class when its kernel and cokernel lie in the class.

The class of all finitely generated abelian groups has the closure properties needed for the Hurewicz theorem modulo a Serre class. In a simply connected space, degreewise membership of homotopy groups and integral homology groups in this class is equivalent.

An abelian group in which no nonzero element has finite order. Free abelian groups are torsion-free, but torsion-free groups need not be free, as the additive rationals show. This algebraic meaning differs from the torsion-free connection in differential geometry.

For a torsion-free abelian group $G$, its rational divisible hull is $G{\otimes }_{\mathbb{Z}}\mathbb{Q}$. It consists of fractions $g/n$, with $n>0$ and $g/n=h/m$ exactly when $mg=nh$. The natural map $G\to G\otimes \mathbb{Q}$ is injective. Every homomorphism from $G$ into a torsion-free divisible Abelian group extends uniquely through this hull.

An abelian subgroup is a subgroup that is an abelian group under the restricted operation.

A finitely generated abelian group has a finite generating set. Equivalently, it is the cokernel of a homomorphism between finitely generated free abelian groups.

The rank of a finitely generated abelian group $G$ is the number $r$ of infinite cyclic factors in its invariant-factor decomposition, equivalently ${\dim }_{\mathbb{Q}}(G{\otimes }_{\mathbb{Z}}\mathbb{Q})$.

Every finitely generated abelian group is isomorphic toThe rank $r$ and invariant factors $d_i>1$ are unique.

For a prime $p$, the Prüfer group $C_{p^{\infty }}$ is the union of an ascending chain of cyclic groups of orders $p,p^2,p^3,\dots$. As a $\mathbb{Z}$-module it is Artinian but not Noetherian.

An additive group is an abelian group whose operation is written as addition, with identity $0$ and inverse of $x$ written $-x$.

A nonzero finitely generated subgroup of the additive rational numbers is an infinite cyclic group. Put the generators over a common positive denominator. Their integer numerators generate the ideal given by their greatest common divisor, and Bezout identity expresses that divisor as an integer combination of them. Dividing back by the denominator gives a single generator.

An element $a$ of an abelian group is infinitely divisible when, for every positive integer $n$, there is an element $b$ with $nb=a$. Compactness can create a nonzero infinitely divisible element in a nonstandard model even when the original group has none.

A finite abelian group is a finite group whose operation is commutative.

An elementary abelian $p$-group is a finite abelian group in which every nonidentity element has order $p$. It is equivalently a finite-dimensional vector space over ${\mathbb{F}}_p$ under addition.

Pontryagin duality associates to a locally compact abelian group the group of its continuous characters and identifies the double dual canonically with the original group.

The character group $\hat{G}$ consists of the group homomorphisms $\gamma :G\to {\mathbb{C}}^{\times }$. Pointwise multiplication makes it a finite abelian group isomorphic to $G$.

If $H$ is a subgroup of a finite abelian group, $a\notin H$, and $k$ is the least positive integer with $ka\in H$, then each character of a finite abelian group on $H$ has exactly $k$ extensions to $H+⟨a⟩$. The displayed root choice makes the extension well-defined. Starting with the trivial subgroup and adjoining generators proves that the number of characters equals the group order without using a decomposition into cyclic groups.

A character of a finite abelian group is a homomorphism into the complex unit circle. Equivalently it is a linear character; finite order forces each character value to be a root of unity. These are the elements of the character group of a finite abelian group and form the orthogonal Fourier basis of functions on the group.

For a linear character $\chi$ of a finite group $K$,Indeed, if $\chi (h)\ne 1$, the bijection $k\mapsto hk$ gives ${\sum }_k\chi (k)=\chi (h){\sum }_k\chi (k)$, forcing the sum to vanish. The trivial character gives $|K|$ directly. The same argument applies to the restriction of a linear character of a larger group to a subgroup.

For a subgroup $K$ of a finite abelian group $G$, its annihilator isThese are exactly the linear characters that descend to the quotient group $G/K$. The character group of a finite abelian group has the same size as the group, so $|K^{\perp }|=|G/K|=|G|/|K|$. The annihilator is what abelian hidden-subgroup Fourier sampling measures.

A finite additive group is a finite group whose operation is written as addition. Its identity element is denoted $0$, the inverse element of $x$ is denoted $-x$, and translation $y\mapsto x+y$ is a bijection.

A subgroup is a subset of a group that is itself a group under the restricted operation.

A Hall subgroup has order coprime to its index. For a set of primes $\pi$, a Hall $\pi$-subgroup has order using only primes of $\pi$ and index using only primes outside $\pi$. Thus its order contains the full prime-power contributions belonging to $\pi$. A Sylow subgroup is the single-prime instance. Hall subgroup existence in soluble groups generalizes Sylow existence to every prime set.

Every finite soluble group has a Hall subgroup for each prime set. Induct on the group order using an elementary abelian minimal normal subgroup. In the coprime hard case, lift a minimal normal subgroup of the quotient, choose its Sylow subgroup, and apply the Frattini argument. A proper normalizer reduces the order; a normal Sylow subgroup allows induction in its quotient. This proves existence without assuming an independent complement theorem.

The union of two subgroups is a subgroup exactly when one contains the other. Otherwise choose $h\in H\setminus K$ and $k\in K\setminus H$. Their product cannot lie in $H$, since $h^{-1}(hk)=k$ would then lie in $H$, and cannot lie in $K$, since $(hk)k^{-1}=h$ would then lie in $K$. The union therefore fails closure. Unlike the union, the subgroup generated by both subgroups always exists.

The index $[G:H]$ is the number of left cosets of $H$ in $G$.

Every subgroup $H\le G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.

A subgroup $H\le G$ has finite index when the coset space $G/H$ is finite. Intersecting a finite-index subgroup with a subgroup again gives a finite-index subgroup of the latter.

For a group homomorphism $\varphi :G\to H$, the map $g\ker \varphi \mapsto \varphi (g)$ gives an isomorphism

The direct product $G\times H$ has componentwise multiplication. An element $(g,h)$ of finite component orders has order $\mathrm{lcm}(\mathrm{ord}g,\mathrm{ord}h)$.

The restricted direct sum of a family of groups $G_i$ is the subgroup of their direct product of groups consisting of tuples with only finitely many nonidentity coordinates. Multiplication is coordinatewise. If every factor is a finite group, each subgroup that is a finitely generated group lies in a finite subproduct, so the restricted direct sum is a locally finite group. This gives concrete infinite torsion groups without a uniform bound on element orders.

If $H,K⊴G$, $H\cap K=\{1\}$, and $HK=G$, then multiplication defines an isomorphism $H\times K\to G$. Conversely, the two canonical factors of a direct product satisfy these conditions.

A group is metacyclic when it has a cyclic normal subgroup whose quotient is cyclic. Every dihedral group is metacyclic through its rotation subgroup.

The commutator subgroup $G^{\prime }$ is generated by all $[x,y]=x^{-1}{y}^{-1}xy$. It is characteristic, $G/G^{\prime }$ is abelian, and every normal subgroup with abelian quotient contains $G^{\prime }$.

A group $G$ is perfect when $G^{\prime }=G$, equivalently when its abelianization is trivial.

The abelianization of a group is the quotient $G^{\mathrm{ab}}=G/G^{\prime }$ by its commutator subgroup. It is the largest abelian quotient of $G$.

For odd $n\ge 3$, the dihedral group $D_{2n}$ of order $2n$ has rotation generator $r$ and reflection generator $s$, with $srs=r^{-1}$. The group commutator $[r,s]=r^{-2}$ generates the rotation subgroup, while the quotient by that subgroup is $C_2$. Hence the commutator subgroup is exactly $⟨r⟩$ and the abelianization is $C_2$. The only abelian quotients are $C_2$ and the trivial group.

The centre of a group is $Z(G)=\{z\in G:zg=gz\text{ for every }g\in G\}$. It is a normal subgroup. A nontrivial finite p-group has nontrivial centre: in the class equation, each noncentral conjugacy class has size divisible by $p$, so $|Z(G)|\equiv |G|\equiv 0(\mathrm{mod}p)$.

If $G/Z(G)$ is cyclic, then $G$ is abelian. Indeed, if $gZ(G)$ generates the quotient, every element is $g^{a}z$ with $z\in Z(G)$, and any two such elements commute.

For a group acting on a ring by ring automorphisms, the elements fixed by every group element form a subring containing $1$. For finite $G$ acting on a commutative ring, each $r$ satisfies the monic orbit equation ${\prod }_{g\in G}(T-g(r))$, whose coefficients are fixed. Thus $R$ is integral over its invariant subring. This uses an orbit product, not division by the group order.

A fraction of invariant elements is fixed by every automorphism. If that fraction is integral over $R^G$, its monic equation also makes it integral over $R$. A normal domain therefore places it in $R$, and fixedness places it in $R^G$. This proves integral closedness of the invariant subring in its own fraction field. The proof works for any group and in every characteristic; no averaging or finiteness of the group is needed for normality.

A transitive group action is primitive if the only blocks are singletons and the whole set; a block has every translate either equal to it or disjoint from it. Orbits of a normal subgroup form blocks. Hence a nontrivial normal subgroup in a faithful primitive action is transitive. Two-transitivity implies primitivity. This normal-subgroup observation is a hypothesis engine for Iwasawa's simplicity lemma.

The symmetric group acts on multivariate polynomials by replacing each variable $x_i$ by $x_{\sigma (i)}$. With $(\sigma \tau )(i)=\sigma (\tau (i))$, substitution gives $\sigma \cdot (\tau \cdot f)=(\sigma \tau )\cdot f$. Thus it is a left group action. A polynomial encoding a combinatorial partition has a group orbit encoding all relabelings and a stabiliser subgroup encoding its label symmetries.

A fixed point of a group action is an element fixed by every group element. The set $A^G=\{a:g\cdot a=a\text{ for all }g\in G\}$ is the categorical limit of the action viewed as a functor from the one-object category associated with $G$ to the Category of sets.

The left regular action of a group on its own underlying set is left multiplication. It is faithful, because an element fixing the identity is itself the identity. For a finite group it gives the permutation representation used in Cayley theorem.

A block system for a group action is a partition of the acted-on set such that every group element permutes its parts. For a transitive action, a nontrivial block system has more than one part and more than one element per part. The full symmetric group preserves no such partition: a transposition exchanging one point in each of two parts breaks it.

A group action on a set $X$ is transitive when every two points lie in the same orbit of a group action, equivalently when for all $x,y\in X$ there is a group element $g$ with $gx=y$.

A regular permutation subgroup acts transitively with trivial point stabilizers. Each ordered pair of points is related by exactly one subgroup element. A finite regular subgroup on $n$ points has order $n$. Normality and characteristicity are additional properties; they do not follow from regularity alone.

An action is sharply $t$-transitive if each ordered $t$-tuple of distinct points is carried to any other by exactly one group element. For finite degree $n\ge t$, its order is $n(n-1)\cdots (n-t+1)$ and each ordered-tuple stabilizer is trivial. Sharp one-transitivity is a regular action. Finite sharp two-transitivity gives a regular kernel of a finite sharply two-transitive group.

In a finite sharply two-transitive group, the identity together with all fixed-point-free elements forms a regular normal subgroup. One proof constructs virtual characters from induced point-stabilizer characters and the augmentation character, proves their norms are one, and realizes this set as an intersection of representation kernels. Conjugation by the point stabilizer is transitive on the kernel's nonidentity elements. This forces the kernel to be elementary abelian, so it is a unique Sylow subgroup and is characteristic. Closure of the fixed-point-free set is a theorem, not something supplied merely by counting its size.

Adjoin a new point $\omega$ fixed by a transitive group $G$. A one-point extension is a transitive group on the enlarged set whose stabilizer of $\omega$ is exactly $G$, acting on the remaining points in the prescribed way. Its order is $(n+1)|G|$ in the finite degree-$n$ case. The double-coset criterion for a one-point extension gives a concrete generator test.

If $x$ interchanges the new point and $\alpha$, put $H=G_{\alpha }$. Then $⟨G,x⟩$ is a one-point extension exactly when $x^2\in H$, $xH{x}^{-1}=H$, and $xgx\in GxG$ for every $g\notin H$. These conditions make $G\cup GxG$ multiplication-closed, hence a group in the finite setting, with new-point stabilizer $G$. Necessity comes from the two double cosets in the extended two-transitive action.

A transitive subgroup $G\le S_p$ for prime $p$ that contains a transposition is $S_p$. Form a graph on the $p$ letters with edges given by all conjugates in $G$ of that transposition. The transitive action permutes connected components, so their common size divides $p$. An edge rules out size one, hence the graph is connected. Its edge transpositions generate $S_p$: successive swaps along a path generate the swap of its endpoint with any chosen start. All these edge transpositions belong to $G$.

A group action on a set is two-transitive when it acts transitively on ordered pairs of distinct points. Equivalently, a point stabilizer acts transitively on the remaining points.

For a group $G$ acting on a set $X$ and a point $x\in X$, the orbit map is $G\to X$, $g\mapsto gx$. Its image is the orbit of a group action of $x$, and its fibres are cosets of the stabilizer subgroup of $x$.

For $G$-sets $X$ and $Y$, a map $f:X\to Y$ is equivariant when $f(gx)=gf(x)$ for every $g\in G$ and $x\in X$.

A group acts on itself and on its subgroups by conjugation: $g\cdot x=gx{g}^{-1}$ and $g\cdot H=gH{g}^{-1}$.

Conjugation by $g$ is the group automorphism $x\mapsto gx{g}^{-1}$. Its orbits are the conjugacy classes.