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For a group action of a group $G$ on a set $X$, the stabilizer subgroup of $x\in X$ consists of the transformations that leave $x$ fixed:It is a subgroup: the identity element fixes $x$, products of transformations fixing $x$ fix $x$, and the inverse element of a transformation fixing $x$ also fixes $x$. A stabilizer subgroup is a subset of $G$, whereas the orbit of a group action is a subset of $X$. See Ackerman's algebra notes, definition 0.4.1.2.

For the square group actions defined under group action, direct calculation givesThe reflection $s$ fixes $A$ and $C$ individually. The reflection $rs$ preserves the unoriented side of a polygon $AB$ by exchanging its two vertices of a polygon. The half-turn rotation $r^2$ exchanges the two ends of the unoriented diagonal of a polygon $AC$ while preserving that diagonal of a polygon. Thus fixing a side of a polygon or diagonal of a polygon as an object does not mean fixing every point on it; the latter condition uses a pointwise stabilizer. The four stabilizer subgroups have group orders $2,2,4,8$, matching group orbit sizes $4,4,2,1$ through the orbit-stabilizer theorem.

A stabilizer subgroup records the transformations still available after imposing a constraint. Fixing a vertex of a polygon of the square leaves two choices; fixing its centroid leaves all eight. In a faithful group action, a stabilizer subgroup can still contain a nonidentity transformation, as $s\in (D_4{)}_A$ shows: a faithful group action requires that only the identity element fix every point of $X$ simultaneously, not that every individual stabilizer subgroup be trivial. The two-point cyclic group example under group action has both stabilizer subgroups equal to the group kernel $\{0,2\}$ of its associated group homomorphism.

A group action is faithful when only the identity acts as the identity on every point.

The full isometry group of a cube has order $48$. Its transitive actions on the $8$ vertices, $12$ edges, and $4$ main diagonals have stabilizers isomorphic respectively toThe action on main diagonals has kernel $\{I,-I\}$.

For a group action of any group $G$ on a set $X$, the cardinality of the orbit of a group action through $x$ is the index of a subgroup of its stabilizer subgroup:In particular, if $G$ is a finite group,For a fixed finite group, a larger stabilizer subgroup means a smaller group orbit: more transformations repeat the same image of $x$. The set $X$ itself need not be finite.

The underlying bijection uses left cosets, without requiring $G_x$ to be a normal subgroup:To check it briefly, replacing $g$ by $gh$ with $h\in G_x$ preserves $g\cdot x$, so $\Phi$ is well-defined. Every member of the group orbit has this form. Finally, $g\cdot x=k\cdot x$ implies $k^{-1}g\in G_x$, hence $g{G}_x=k{G}_x$, proving the bijection. This is the construction in MIT's group-action notes, lemma 23.5. For infinite groups, the bijection and the cardinality statement $|Gx|=[G:G_x]$ still hold; division of infinite cardinalities does not supply the finite formula.

For the eight-element dihedral group $D_4$ acting on the square described under group action, moving a marked vertex of a polygon gives $8=4\cdot 2$. Moving a marked side of a polygon also gives $8=4\cdot 2$; moving one unoriented diagonal of a polygon gives $8=2\cdot 4$; moving the centroid gives $8=1\cdot 8$. These are four different group actions, each with one group orbit. In the first group action, the four left cosets of $H=G_A=\{e,s\}$ areThey correspond respectively to $A,B,C,D$. Counting all eight transformations would count each position of the marked vertex of a polygon twice.

For a second computed example, let the symmetric group $S_3$ act on the Cartesian product $X=\{1,2,3{\}}^2$ by $\sigma \cdot (i,j)=(\sigma (i),\sigma (j))$. There are two orbits of a group action:andThe first stabilizer subgroup can exchange $2$ and $3$ while fixing $1$; the second must fix both $1$ and $2$, so also fixes $3$. The orbit-stabilizer theorem gives $6=3\cdot 2$ and $6=6\cdot 1$, and the two group orbit sizes account for all nine ordered pairs. The general reason for exactly these two group orbits is equality-pattern orbits of ordered pairs.

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

A subgroup $H\le G$ is a subset of a group $G$ that is itself a group under the same group operation. It must contain the identity element of $G$ and be closed under products and inverse elements. Associativity is inherited from $G$, so it needs no new proof. The whole group and the trivial group $\{e\}$ are always subgroups. A proper subgroup satisfies $H\ne G$.

A useful single-condition test isTo see why it works, choose $a\in H$; then $a{a}^{-1}=e\in H$. Taking $a=e$ gives every inverse element $b^{-1}\in H$. Applying the test to $a$ and $b^{-1}$ then gives $ab\in H$. Conversely, every subgroup satisfies the test by closure under products and inverse elements. This also makes the additive version memorable: a nonempty subset of an additive group is a subgroup exactly when it is closed under differences. See McKernan's MIT lecture 2, Proposition 2.3, for the equivalent closure criterion.

For the square dihedral group $D_4$ above, $R=\{e,r,r^2,r^3\}$ is a cyclic subgroup: products add exponents modulo four, and $(r^k{)}^{-1}=r^{-k}$. The smaller collections $\{e,r^2\}$ and $\{e,s\}$ are subgroups of group order two. But $\{e,r\}$ is not a subgroup, since $r\cdot r=r^2$ lies outside it. A plausible-looking list of symmetries therefore need not constitute a complete collection of reversible operations.

Inside $(\mathbb{Z},+)$, the even numbers $2\mathbb{Z}$ form a subgroup: $2m-2n=2(m-n)$ is an even number. Its two cosets consist of the even numbers and the odd numbers, so its index of a subgroup is two. The parity group homomorphism $k\mapsto k\mathrm{mod}2$ has $2\mathbb{Z}$ as its kernel of a group homomorphism. A subgroup can thus identify exactly which information a composition-preserving map forgets.

Subgroups also describe residual symmetry. In the group action of $D_4$ on the four vertices, the stabilizer subgroup of $(1,1)$ is $\{e,sr\}$: the diagonal reflection $sr$ fixes that vertex, and the other six motions move it. Requiring a feature to stay fixed still leaves a group, because both composition and reversal preserve that requirement. This is why subgroups are useful for studying a group action one feature at a time.

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.

An abelian group is a group whose group operation is commutative: $gh=hg$ for every $g,h\in G$. Thus rearranging a finite sequence of operations preserves the result. Associativity already permits changing parentheses; commutativity additionally permits exchanging the elements.

The additive integers, the additive real numbers, and every cyclic group are abelian groups. Additive notation is often convenient: the identity element is written $0$, the inverse element of $a$ is $-a$, and $a+b=b+a$. It does not force an element to be an ordinary number: in $C_2\times C_2$, an element is a pair of binary coordinates, added componentwise modulo two.

Commutativity is useful because an integer linear combination of chosen group elements depends only on their coefficients, rather than on the order in which the operations were performed. The Fundamental theorem of finitely generated abelian groups turns this simplicity into a classification by cyclic groups. However, commutativity does not imply that a single generator of a group suffices: the Klein four-group is the smallest counterexample, as the cyclic group of order four and Klein four-group demonstrate.

A cyclic group is a group that can be generated by repeating a single group element $g$ and its inverse element:For an additive group, the same expression is written $\{kg:k\in \mathbb{Z}\}$: $kg$ means repeated addition, including subtraction when $k$ is negative. A finite cyclic group has a repeating cycle, while an infinite cyclic group is isomorphic to the additive integers. The notation $C_n$ refers to a finite cyclic group of order $n$.

This definition compresses a whole composition rule into one repeatable operation. Once $g$ is chosen, $g^a{g}^b=g^{a+b}$, so working with group elements becomes working with integer exponents. In the finite case those exponents are reduced using modular arithmetic. It also proves that every cyclic group is an abelian group, because $a+b=b+a$. A finite cyclic group of order $n$ has exactly $d$ solutions of $x^d=e$ for every positive integer divisor $d$ of $n$: in exponents, $n\mid dk$ has exactly $d$ solutions modulo $n$. The standard finite and infinite models are described in Group Theory (v4.01) by James S. Milne (section 1.16).

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

Both translations are bijections because the inverse element gives the inverse multiplication operation.

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

An inverse element of $a$ in a monoid is an element $b$ satisfying $a\ast b=b\ast a=e$, where $e$ is the identity element. Its notation is $a^{-1}$. Some elements of a monoid can lack an inverse element; a group requires one for every element. Associativity makes a two-sided inverse element unique: if $b,c$ both invert $a$, then $b=b\ast (a\ast c)=(b\ast a)\ast c=c$.

For the full transformation monoid, a self-map has an inverse element exactly when it is a bijection; that element is its inverse function. On $\{0,1\}$, the swap $s(0)=1,s(1)=0$ is its own inverse element, while a constant function has none because it merges the two inputs.

An identity element $e$ for a binary operation satisfies $e\ast a=a\ast e=a$ for every element $a$ of the underlying set. It is a two-sided identity: merely satisfying one of the two equations is insufficient. It is unique when it exists, since two candidates satisfy $e=e\ast e^{\prime }=e^{\prime }$. This uniqueness does not require the associative property.

Zero is the identity element for addition of integers, one for multiplication of integers, and the identity function for function composition of self-maps. The empty word is the identity element for concatenation. An associative operation with an identity element makes a monoid; inverse elements are an additional requirement for a group.

A nontrivial group has more than one element.

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.

Green's third identity represents a function inside a domain through its boundary values, normal derivatives, and a volume term involving its Laplacian.

Two integers $a,b$ are coprime when their greatest common divisor is one:For positive integers, this says that the only positive common divisor is one. For signed integers, use $\gcd (a,b)=\gcd (|a|,|b|)$, with absolute values; changing either sign leaves coprimality unchanged. With the convention $\gcd (0,0)=0$, zero is coprime precisely to $1,-1$, because $\gcd (a,0)=|a|$. Both $1,-1$ are coprime to every integer. ^[ref]

The word concerns shared prime factors, not whether either input is a prime number. For instance $8=2^3$ and $9=3^2$ are coprime composite numbers, even though neither is a square-free integer. Conversely, $6=2\cdot 3$ and $10=2\cdot 5$ are square-free integers but are not coprime. Consecutive integers are always coprime: any common divisor of $n,n+1$ divides their difference one. Thus $14,15$ provide another pair of coprime composite numbers.

For positive integers, let $P(n)$ be the prime support of an integer. The equivalent test isThe set intersection of the two prime supports of an integer must be the empty set. Indeed, any positive common divisor greater than one has a prime factor, and any shared prime factor is a common divisor greater than one. Hence coprimality depends on which prime factors occur, rather than on their prime valuations; replacing either input by its radical of an integer preserves coprimality.

For a list of at least two integers, distinguish pairwise coprimality, requiring every pair to be coprime, from joint coprimality, requiring only the greatest common divisor of the entire list to be one. For example,No prime factor belongs to all three prime supports of an integer, yet each pair shares one. By contrast, $4,9,25$ have pairwise coprimality. The distinction is presented by Poritz under the names pairwise and mutually relatively prime. ^[ref]

One use is simplifying fractions. For a fraction $a/b$ with integer entries and $b>0$, divide its numerator and denominator by $d=\gcd (a,b)$ to obtain a reduced fraction with coprime entries. For example, $42/30=7/5$ since $d=6$, and $-42/30=-7/5$ uses the same $d$. A further common divisor would make $d$ too small, so the reduction is complete. In particular, the reduced fraction for zero is $0/1$.

Another use is reversible modular arithmetic: for an integer modulus $m>1$, an integer $a$ has a modular inverse precisely when $a,m$ are coprime. The proof outline uses Bézout's identity: $ua+vm=1$ makes $u$ a modular inverse, and any inverse relation $ua-km=1$ forces every common divisor to divide one. For example, $3\cdot 5-2\cdot 7=1$ gives $5^{-1}\equiv 3(\mathrm{mod}7)$, whereas $6$ has no modular inverse modulo $10$. Thus coprimality is the exact condition allowing division by a residue in modular arithmetic. ^[ref]

The shearing sheet is a local Cartesian approximation to a differentially rotating disk. It retains Coriolis force, linearized tidal gravity, and a background linear shear.

The real Grassmannian of $k$-planes is represented by symmetric idempotent matrices of trace $k$.

A linear map between Banach spaces is continuous if and only if its graph is closed.