Glossary

The group of rotations and reflections that leave a molecule's shape unchanged while fixing at least one point in space -- water's is C_{2v}, ammonia's is C_{3v}.

A group in which the operation commutes: a*b = b*a for every pair of elements.

G^{\text{ab}}=G/[G,G], the largest abelian quotient of G -- always abelian, no matter how non-abelian G itself is.

A set together with one or more operations defined on it, studied for the properties it has rather than for what its elements are.

The basic operations of adding, subtracting, multiplying, and dividing numbers.

An operation is associative when (a*b)*c = a*(b*c) for all elements -- grouping never changes the result.

An isomorphism from a group G to itself. The set of all automorphisms of G, written \operatorname{Aut}(G), forms a group under composition.

A pairing between two sets that matches every element of one to exactly one element of the other, with none left out or shared.

A rule that combines two elements of a set to produce a single element of that same set: a function G \times G \to G.

An arrangement of N colored beads around a circle, where two arrangements count as the same when either a rotation or a reflection carries one to the other.

A graph built from a group G and a generating set S: one vertex per element of G, and one directed edge x \to xs for every vertex x and every s \in S.

A square table listing every operation's result, rows and columns both indexed by the set's elements.

The set Z(G) of elements that commute with every element of G -- always a normal subgroup.

A subgroup fixed setwise by every automorphism of G, not just by conjugation -- a strictly stronger condition than normal.

A complete list of every possible structure of a given kind, up to relabeling.

A set is closed under an operation when combining any two of its elements always produces another element already in the set.

[G,G], the subgroup generated by every commutator in G -- trivial exactly when G is abelian.

One of the simple quotient groups H_{i+1}/H_i appearing in a composition series.

A subnormal series in which every quotient H_{i+1}/H_i is simple.

Two integers a and b are congruent mod n, written a \equiv b \pmod n, when n divides their difference.

For h\in G, the orbit \operatorname{Cl}(h)=\{ghg^{-1}:g\in G\} of h under G acting on itself by conjugation.

For g,h in a group, the element ghg^{-1} -- the result of shifting into g's viewpoint, applying h, then shifting back.

The act of transforming an element by sandwiching it between another element and its inverse: g^{-1}xg.

Any single element of a coset, since any element of gH can be used to name that same coset: x \in gH \implies xH = gH.

H is covered by K when H < K and no subgroup of the group sits strictly between them -- drawn as a direct edge in a subgroup lattice.

A permutation written as a product of cycles, each cycle listing the elements it moves in the order it moves them, e.g. (1\,3\,2).

A group in which every element is a power of a single element -- that one element generates the whole group.

The sequence G^{(0)}=G,\ G^{(i+1)}=[G^{(i)},G^{(i)}], built by repeated commutator subgroups.

The group of all 2n symmetries of a regular n-gon -- n rotations and n reflections -- under composition.

Two or more cycles that share no elements -- every permutation decomposes into a unique set of disjoint cycles, up to the order they're listed in and which element starts each one.

A procedure for computing \gcd(a,b) by repeated division: replace (a,b) with (b,r), the divisor and remainder from dividing a by b, until the remainder reaches 0 -- the last nonzero remainder is the gcd.

\varphi(n) counts the integers from 1 to n that are relatively prime to n -- the order of U(n).

The small raised number showing how many times to repeat an operation, like a^3 = a \cdot a \cdot a.

The set of ordered pairs (g,h) with g\in G, h\in H, combined componentwise -- |G\times H|=|G||H| when finite.

A group action in which the only element fixing every point of X is e itself -- the kernel \bigcap_{x\in X}\operatorname{Stab}(x) is trivial.

For a nonempty finite subset H of a group G, closed under the operation, H is automatically a subgroup -- closure alone forces the identity and every inverse to already be present.

A group with some FINITE subset generating all of it, regardless of whether the group itself is finite.

A group admitting a presentation \langle S \mid R\rangle with both the generating set S and the relation list R finite.

For g \in G acting on a set X, a point x \in X with g\cdot x = x; the set of all such points is \text{Fix}(g).

The group F(S) of all reduced words in a generating set S, where the only equalities among words are those forced by bare cancellation.

The symmetry group of a pattern repeating in exactly one direction, such as a decorative border -- exactly seven of them exist, up to the type of symmetry each one carries.

The smallest subgroup containing a given subset, written \langle S \rangle -- every element built by combining the subset's elements and their inverses.

An element g of a cyclic group whose powers g^0, g^1, g^2, \ldots produce every element of the group.

A matrix whose rows are chosen codewords spanning an entire linear code -- every codeword is some combination of its rows, mod 2.

The largest positive integer dividing both of two given integers, written \gcd(a,b).

A set G with a binary operation satisfying closure, associativity, an identity element, and inverses for every element.

A function G\times X\to X, written g\cdot x, satisfying e\cdot x=x and g\cdot(h\cdot x)=(gh)\cdot x for every g,h\in G and x\in X.

A group G built from a normal subgroup N and a quotient G/N \cong Q -- "an extension of Q by N" -- generalizing the semidirect product to allow N to sit inside G in a way that need not split apart.

A function f: G \to H between groups with f(a*b) = f(a)\bullet f(b) for every a,b \in G -- combining first or mapping first always gives the same answer.

An element e that leaves every other element unchanged when combined with it: e*a = a*e = a.

For a homomorphism f: G \to H, the set \operatorname{im}(f) = \{f(g) : g \in G\} -- always a subgroup of H, though never automatically normal in H.

Written [G:H], the number of distinct left cosets of a subgroup H in G -- equivalently |G|/|H| for a finite group, by Lagrange's theorem.

For a group element a and integer n, the result a^n of combining a with itself n times (n>0), its inverse n times (n<0), or the identity (n=0).

The set of congruence classes \{0, 1, \ldots, n-1\} under addition mod n -- a group of order n.

A group G with normal subgroups H,K satisfying H\cap K=\{e\} and HK=G -- equivalently, G\cong H\times K.

For subgroups H, K \le G, the set H \cap K of elements common to both -- always itself a subgroup of G.

For an element a, an element a^{-1} with a*a^{-1} = a^{-1}*a = e, the identity.

A bijective homomorphism f: G \to H; when one exists, G and H are called isomorphic, written G \cong H, meaning they are structurally identical.

For a homomorphism f: G \to H, the set \ker(f) = \{g \in G : f(g) = e_H\} -- always a normal subgroup of G.

For H \le G and g \in G, the set gH = \{gh : h \in H\} -- every element of H multiplied by g on the left.

A set of codewords in (\mathbb{F}_2)^n that is closed under bitwise addition (XOR) -- equivalently, a subgroup of (\mathbb{F}_2)^n.

Computing a^b \bmod n -- raising a to the power b and reducing mod n, typically by squaring and reducing repeatedly rather than forming the full power first.

An arrangement of N colored beads around a circle, where two arrangements count as the same exactly when a rotation carries one to the other.

The number of steps a nilpotent group's upper central series takes to reach the whole group.

Every continuous symmetry of a physical system's laws corresponds to a conserved quantity -- rotational symmetry to angular momentum, time symmetry to energy.

A group with at least one pair of elements that does not commute under the operation.

A subgroup H \le G with gHg^{-1}=H for every g \in G -- equivalently, gH=Hg for every g.

The set N_G(H) of elements g \in G with gHg^{-1}=H -- always satisfies H \le N_G(H) \le G, with H \trianglelefteq G exactly when N_G(H)=G.

For a group G acting on X and a point x\in X, the set \operatorname{Orb}(x)=\{g\cdot x:g\in G\} of every place x can be sent.

The number of elements in a group, written |G| -- finite if that count is finite, infinite otherwise.

The smallest positive integer n with a^n = e, if one exists; the element has infinite order otherwise.

An automorphism that is not conjugation by any element of the group -- D_4 has some, S_3 has none.

A finite group whose order is p^n for a single prime p and some n\ge0 -- equivalently, a group in which every element's order is a power of p.

Whether a permutation decomposes into an even or an odd number of transpositions -- every decomposition of the same permutation shares one parity, so it's well-defined.

A way of splitting a set into pieces that don't overlap and together cover the whole set.

A group G with [G,G]=G itself -- trivial abelianization, the opposite extreme from abelian. A_5 is the smallest nontrivial example.

A bijection from a finite set to itself -- a rearrangement of the set's elements.

A compact recipe for a group: a list of generators plus the relations they must satisfy.

For subgroups H, K \le G, the set HK = \{hk : h \in H, k \in K\} -- a subgroup exactly when HK=KH, not automatically.

A refinement of Burnside's lemma that tracks which specific labels a configuration uses, via a symbolic color-tracking polynomial, rather than only a bare total count.

The set of all cosets of a normal subgroup N in G, written G/N, made into a group by (g_1N)(g_2N) = (g_1g_2)N.

A word with no symbol sitting immediately next to its own inverse -- nothing left to cancel.

A rigid motion that flips a figure across a line (or plane), like a mirror image.

A graph in which every vertex has the same degree.

An equation that says how a group's generators interact -- part of the recipe that defines the group.

Two integers whose only common positive divisor is 1, written \gcd(a,b)=1.

For H \le G and g \in G, the set Hg = \{hg : h \in H\} -- every element of H multiplied by g on the right. Need not equal the matching left coset gH unless G is abelian or H is normal.

A rigid motion that turns a figure around a fixed point or axis, without flipping or sliding it.

If a normal subgroup's order and its quotient's order share no common factor, the extension always splits -- no coprime-order extension can be a genuinely tangled, non-splitting one.

A generalization of the direct product where H's action on N (by conjugation) may be non-trivial -- the direct product is the special case where the action is trivial.

A nontrivial group whose only normal subgroups are \{e\} and itself -- no proper nontrivial normal subgroup exists to break it down further.

A group whose derived series reaches the trivial subgroup after finitely many steps.

For a group G acting on X and a point x\in X, the subgroup \operatorname{Stab}(x)=\{g\in G:g\cdot x=x\} of elements leaving x fixed.

A directed graph in which every vertex can reach every other vertex by following arrows forward -- the directed-graph strengthening of ordinary connectedness.

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

A Hasse diagram showing every subgroup of a group, arranged by rank, with an edge between H and K exactly when H is covered by K.

A chain \{e\}=H_0 \trianglelefteq H_1 \trianglelefteq \cdots \trianglelefteq H_n = G where each H_i is normal in H_{i+1} specifically -- not necessarily normal in G itself.

For |G|=p^am with p\nmid m, a subgroup of order exactly p^a -- the full p-share of |G|. \operatorname{Syl}_p(G) denotes the set of all of them; n_p=|\operatorname{Syl}_p(G)| denotes how many there are.

A rigid motion (rotation or reflection) that maps a figure onto itself, leaving its appearance unchanged.

Checking a received word against a parity-check matrix to get a syndrome vector -- zero means no detected error, and a nonzero pattern points at which bit most likely flipped.

The subgroup of S_n consisting of every even permutation, of order n!/2 for n \ge 2.

The set of units modulo n, under multiplication mod n -- a group of order \varphi(n).

The group of all n! permutations of \{1,\ldots,n\} under composition.

A group action in which \operatorname{Orb}(x)=X for every point x -- a single orbit covers the entire set.

A 2-cycle -- a permutation that swaps exactly two elements and fixes everything else.

The subgroup containing only the identity element, \{e\} -- present in every group.

A permutation written as a 2 \times n array, top row the inputs in order and bottom row each input's image directly beneath it.

An integer a with 1 \le a < n that has a multiplicative inverse mod n -- equivalently, \gcd(a,n)=1.

The sequence Z_0=\{e\},\ Z_1=Z(G),\ Z_{i+1}/Z_i=Z(G/Z_i), built bottom-up from repeated centers.

A single point or node in a graph or geometric figure, where edges or sides meet.

The symmetry group of a pattern repeating in two independent directions across the plane -- exactly seventeen exist, the two-dimensional counterpart of the seven frieze groups.

An operation defined in terms of representatives is well-defined when its result never depends on which representative was chosen.