Groups

Introduction

This article introduces the theory of groups, the algebraic structures that formalize the notion of symmetry. A group is a set equipped with a single binary operation that is associative, has an identity element, and in which every element is invertible. Groups describe the symmetries of geometric figures, the permutations of a finite set, the invertible linear transformations of a vector space, and the units of a ring or algebra.

The treatment is introductory and purely mathematical. We assume familiarity with sets, functions, and the basic theory of rings and algebras from Rings: A General Introduction and Algebras: A General Introduction. Rings carry two operations and algebras add a product; a group carries one. Every ring $(R, +, \cdot)$ contains the additive group $(R, +)$ and the group of units $(R^\times, \cdot)$, and every unital associative algebra $A$ has a group of units $A^\times$.

We begin with the definition and elementary properties of groups, then develop the fundamental constructions: subgroups, homomorphisms, cosets, quotient groups, and group actions. We then study the standard classes of groups (cyclic, symmetric, alternating, dihedral, matrix, and quaternion group) and the two structure theorems that organize the theory, the Sylow theorems for finite groups and the classification of finitely generated abelian groups.

Part I: Groups and Their Elements

1. Definition of a Group

A group is a set $G$ equipped with a binary operation

$$ G \times G \to G, \qquad (a, b) \mapsto a b, $$

called multiplication or composition, satisfying the following axioms.

(G1) Associativity. $(a b) c = a (b c)$ for all $a, b, c \in G$.

(G2) Identity. There is an element $e \in G$ such that $e a = a e = a$ for all $a \in G$.

(G3) Inverses. For every $a \in G$ there is an element $a^{-1} \in G$ with $a a^{-1} = a^{-1} a = e$.

A group is abelian (or commutative) if $a b = b a$ for all $a, b \in G$; in that case the operation is often written additively as $a + b$, with identity $0$ and inverse $-a$. The order of a group, written $|G|$, is the cardinality of its underlying set, and $G$ is finite if $|G|$ is finite.

Examples. The additive groups $(\mathbb{Z}, +)$ and $(\mathbb{Z}/n\mathbb{Z}, +)$, the latter of order $n$, are abelian; $\mathbb{Z} \setminus \{0\}$ is not a group under multiplication, since only $\pm 1$ are invertible in $\mathbb{Z}$. The symmetric group $S_n$ of bijections of $\{1, \ldots, n\}$ has order $n!$ and is nonabelian for $n \geq 3$, and the general linear group $GL_n(F)$ of invertible matrices over a field $F$ is a group. For any ring $R$, the units $R^\times$ form a group.

2. Elementary Properties

Let $G$ be a group.

(a) The identity element is unique.

(b) Inverses are unique: if $a b = b a = e$ and $a c = c a = e$, then $b = e b = (c a) b = c (a b) = c$.

(c) $(a^{-1})^{-1} = a$ and $e^{-1} = e$.

(d) $(a b)^{-1} = b^{-1} a^{-1}$, and more generally $(a_1 \cdots a_n)^{-1} = a_n^{-1} \cdots a_1^{-1}$.

(e) Cancellation holds: $a b = a c$ implies $b = c$, and $b a = c a$ implies $b = c$. Hence $a x = b$ and $y a = b$ have the unique solutions $x = a^{-1} b$ and $y = b a^{-1}$.

3. Order, Powers, and Cyclic Subgroups

For $a \in G$ define $a^0 = e$, $a^{n+1} = a^n a$ for $n \geq 0$, and $a^{-n} = (a^{-1})^n$ for $n \geq 1$. Then

$$ a^{m+n} = a^m a^n, \qquad (a^m)^n = a^{m n} $$

for all $m, n \in \mathbb{Z}$. If $a$ and $b$ commute, then $(a b)^n = a^n b^n$ for all $n$; without commutativity this fails.

The order of $a \in G$ is the least positive integer $n$ with $a^n = e$, if one exists, and is $\infty$ otherwise. The cyclic subgroup generated by $a$ is

$$ \langle a \rangle = \{a^n : n \in \mathbb{Z}\}, $$

and its order equals the order of $a$. A group $G$ is cyclic if $G = \langle a \rangle$ for some $a \in G$. We write $C_n$ for a cyclic group of order $n$.

Examples. In $\mathbb{Z}/n\mathbb{Z}$ the class of $k$ has order $n / \gcd(n, k)$. In $S_n$ the order of a permutation is the least common multiple of the lengths of its disjoint cycles.

Part II: Subgroups, Homomorphisms, and Quotients

4. Subgroups

A nonempty subset $H \subseteq G$ is a subgroup if $a b^{-1} \in H$ for all $a, b \in H$. Equivalently, $H$ is closed under multiplication and inversion and contains $e$. A subgroup is itself a group under the inherited operation.

Examples. The trivial subgroup $\{e\}$ and $G$ itself; $n \mathbb{Z} \subseteq \mathbb{Z}$; the special linear group $SL_n(F) = \{A : \det A = 1\}$; and any cyclic subgroup $\langle a \rangle$. Intersections of subgroups are subgroups, but unions generally are not.

The center of $G$ is

$$ Z(G) = \{z \in G : z g = g z \text{ for all } g \in G\}, $$

an abelian subgroup. For a subset $S \subseteq G$, the subgroup generated by $S$, written $\langle S \rangle$, is the smallest subgroup containing $S$; it consists of all finite products of elements of $S$ and their inverses. The group is finitely generated if some finite $S$ generates it.

5. Group Homomorphisms

Let $G$ and $H$ be groups. A group homomorphism is a function $\varphi : G \to H$ with

$$ \varphi(a b) = \varphi(a) \varphi(b) $$

for all $a, b \in G$. It follows that $\varphi(e_G) = e_H$ and $\varphi(a^{-1}) = \varphi(a)^{-1}$. A bijective homomorphism is an isomorphism, and we write $G \cong H$ when one exists. An isomorphism $G \to G$ is an automorphism, and the automorphisms form a group $\operatorname{Aut}(G)$ under composition.

The kernel and image are

$$ \ker \varphi = \{a \in G : \varphi(a) = e_H\}, \qquad \operatorname{im} \varphi = \{\varphi(a) : a \in G\}. $$

The kernel is a subgroup of $G$ and the image is a subgroup of $H$. A homomorphism is injective if and only if $\ker \varphi = \{e_G\}$.

Example. The determinant $\det : GL_n(F) \to F^\times$ is a surjective homomorphism with kernel $SL_n(F)$.

For each $g \in G$, conjugation $c_g(x) = g x g^{-1}$ is an automorphism of $G$. The map $g \mapsto c_g$ is a homomorphism $G \to \operatorname{Aut}(G)$ whose kernel is $Z(G)$ and whose image is the group $\operatorname{Inn}(G)$ of inner automorphisms; hence

$$ \operatorname{Inn}(G) \cong G / Z(G). $$

6. Cosets and Lagrange's Theorem

Let $H \subseteq G$ be a subgroup and $g \in G$. The left coset of $H$ by $g$ is $g H = \{g h : h \in H\}$, and the right coset is $H g = \{h g : h \in H\}$.

The left cosets partition $G$, and $g H = h H$ if and only if $h^{-1} g \in H$; otherwise $g H$ and $h H$ are disjoint. Every left coset has the same cardinality as $H$. The index of $H$ in $G$, written $[G : H]$, is the number of distinct left cosets, which equals the number of right cosets.

Lagrange's theorem. If $G$ is a finite group and $H \subseteq G$ is a subgroup, then

$$ |G| = [G : H] \, |H|. $$

In particular, $|H|$ divides $|G|$.

Corollaries. The order of every element of $G$ divides $|G|$; every group of prime order $p$ is cyclic, isomorphic to $\mathbb{Z}/p\mathbb{Z}$; and $a^{|G|} = e$ for every $a \in G$.

Applying the last corollary to the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$, of order $p - 1$, gives Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$ for $p \nmid a$. Applying it to $(\mathbb{Z}/n\mathbb{Z})^\times$, of order $\varphi(n)$, gives Euler's theorem: $a^{\varphi(n)} \equiv 1 \pmod n$ when $\gcd(a, n) = 1$.

The converse of Lagrange's theorem is false: the group $A_4$ of order $12$ has no subgroup of order $6$.

7. Normal Subgroups and Quotient Groups

A subgroup $N \subseteq G$ is normal, written $N \trianglelefteq G$, if

$$ g N g^{-1} = N $$

for all $g \in G$; equivalently, $g N = N g$ for all $g$. When $N$ is normal, the set of left cosets becomes a group under

$$ (g N)(h N) = (g h) N. $$

This is the quotient group $G/N$, of order $[G : N]$. Normality is exactly the condition that makes this product well-defined: without it the product of cosets depends on the chosen representatives.

Every subgroup of an abelian group is normal, and $Z(G) \trianglelefteq G$. Every subgroup of index $2$ is normal, since left and right cosets coincide; hence $A_n \trianglelefteq S_n$; likewise $SL_n(F) \trianglelefteq GL_n(F)$, being the kernel of the determinant.

A nontrivial group is simple if its only normal subgroups are $\{e\}$ and $G$ itself. Cyclic groups of prime order are simple, and the alternating groups $A_n$ are simple for $n \geq 5$.

8. The Isomorphism Theorems

Let $\varphi : G \to H$ be a group homomorphism. Then $\ker \varphi \trianglelefteq G$, and the following hold.

First isomorphism theorem. $G / \ker \varphi \cong \operatorname{im} \varphi$.

Second isomorphism theorem. If $H \subseteq G$ is a subgroup and $N \trianglelefteq G$, then $H N$ is a subgroup of $G$, $H \cap N \trianglelefteq H$, and

$$ H N / N \cong H / (H \cap N). $$

Third isomorphism theorem. If $N \subseteq M$ are both normal in $G$, then $M/N \trianglelefteq G/N$ and

$$ (G/N) / (M/N) \cong G/M. $$

Correspondence theorem. Subgroups of $G$ containing $N$ correspond bijectively to subgroups of $G/N$, normal subgroups correspond to normal subgroups, and indices are preserved: $[G : H] = [G/N : H/N]$.

9. The Center, the Commutator Subgroup, and Abelianization

For $a, b \in G$, the commutator is $[a, b] = a b a^{-1} b^{-1}$, which equals $e$ exactly when $a$ and $b$ commute. The commutator subgroup (or derived subgroup) $[G, G]$ is the subgroup generated by all commutators. It is normal in $G$, and the quotient

$$ G^{\mathrm{ab}} = G / [G, G] $$

is abelian, called the abelianization of $G$.

Universal property. Every homomorphism from $G$ to an abelian group has $[G, G]$ in its kernel, hence factors uniquely through $G \to G^{\mathrm{ab}}$. Thus $G^{\mathrm{ab}}$ is the largest abelian quotient of $G$.

Examples. For abelian $G$, $[G, G] = \{e\}$. For $n \geq 3$, $[S_n, S_n] = A_n$, so $S_n^{\mathrm{ab}} \cong \mathbb{Z}/2\mathbb{Z}$. For the quaternion group, $[Q_8, Q_8] = \{\pm 1\}$, so $Q_8^{\mathrm{ab}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

Part III: Group Actions

10. Group Actions

An action of a group $G$ on a set $X$ is a function

$$ G \times X \to X, \qquad (g, x) \mapsto g \cdot x, $$

satisfying $e \cdot x = x$ and $(g h) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$. An action is the same data as a homomorphism $\rho : G \to S_X$, called a permutation representation; the action is faithful if $\rho$ is injective.

Cayley's theorem. Left multiplication, $g \cdot x = g x$, is a faithful action of $G$ on itself. Hence $G$ embeds in $S_G$, and every finite group of order $n$ embeds in $S_n$.

Examples. Conjugation $g \cdot x = g x g^{-1}$ on $G$; left multiplication on the set $G/H$ of cosets, $g \cdot (x H) = (g x) H$, which is defined for every subgroup $H$ and is transitive; the natural action of $S_n$ on $\{1, \ldots, n\}$ and on its $k$-element subsets; and the action of $GL_n(F)$ on the nonzero vectors, and on the lines, of $F^n$.

11. Orbits, Stabilizers, and the Orbit–Stabilizer Theorem

Let $G$ act on $X$, and fix $x \in X$. The orbit of $x$ is

$$ \operatorname{Orb}(x) = \{g \cdot x : g \in G\}, $$

and the stabilizer of $x$ is

$$ \operatorname{Stab}(x) = \{g \in G : g \cdot x = x\}, $$

a subgroup of $G$. The relation $x \sim y$ if $y \in \operatorname{Orb}(x)$ is an equivalence relation, so the orbits partition $X$. The action is transitive if it has a single orbit.

Orbit–stabilizer theorem. The map

$$ G / \operatorname{Stab}(x) \to \operatorname{Orb}(x), \qquad g \operatorname{Stab}(x) \mapsto g \cdot x $$

is a well-defined bijection. Hence for a finite group $G$,

$$ |\operatorname{Orb}(x)| = [G : \operatorname{Stab}(x)] = \frac{|G|}{|\operatorname{Stab}(x)|}, $$

so the orbit size divides $|G|$.

Example. The group $S_4$ acts on the six two-element subsets of $\{1,2,3,4\}$. The orbit of $\{1,2\}$ has size $6$ and stabilizer $\{e,(1\,2),(3\,4),(1\,2)(3\,4)\}$ of order $4$; indeed $24 = 6 \cdot 4$.

12. Conjugacy, the Class Equation, and $p$-Groups

The conjugation action of $G$ on itself has orbits called conjugacy classes, which partition $G$. Two elements $x, y$ are conjugate if $y = g x g^{-1}$ for some $g$. The stabilizer of $x$ under conjugation is the centralizer

$$ C_G(x) = \{g \in G : g x g^{-1} = x\} = \{g \in G : g x = x g\}. $$

An element has a singleton conjugacy class exactly when it lies in the center.

Class equation. Let $G$ be finite, and let $x_1, \ldots, x_k$ represent the conjugacy classes of size greater than $1$. Then

$$ |G| = |Z(G)| + \sum_{i=1}^{k} [G : C_G(x_i)], $$

each index being greater than $1$ and dividing $|G|$.

Example. In $S_4$ the class sizes are $1, 3, 6, 6, 8$, so $24 = 1 + 3 + 6 + 6 + 8$, the leading term being $|Z(S_4)| = 1$.

A finite group $G$ with $|G| = p^n$ for a prime $p$ is a $p$-group.

Theorem. Every finite $p$-group has nontrivial center. Indeed, $p$ divides $|G|$ and each index $[G : C_G(x_i)]$, so $p$ divides $|Z(G)|$.

Corollary. Every group of order $p^2$ is abelian, isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$ or to $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$. For if $|Z(G)| = p$, then $G/Z(G)$ would be cyclic of order $p$, which by the standard lemma forces $G$ abelian and hence $Z(G) = G$, a contradiction.

Cauchy's theorem. If a prime $p$ divides $|G|$, then $G$ contains an element of order $p$.

13. The Sylow Theorems

Let $G$ be finite with $|G| = p^k m$, where $p$ is prime and $p \nmid m$. A Sylow $p$-subgroup of $G$ is a subgroup of order $p^k$, the largest power of $p$ dividing $|G|$.

Sylow I (existence). For every prime $p$ dividing $|G|$, the group $G$ has a Sylow $p$-subgroup.

Sylow II (conjugacy). Any two Sylow $p$-subgroups are conjugate, and every $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup.

Sylow III (counting). The number $n_p$ of Sylow $p$-subgroups satisfies

$$ n_p \mid m \qquad \text{and} \qquad n_p \equiv 1 \pmod p. $$

Moreover $n_p = [G : N_G(P)]$ for any Sylow $p$-subgroup $P$, where $N_G(P) = \{g \in G : g P g^{-1} = P\}$. In particular, $P \trianglelefteq G$ if and only if $n_p = 1$.

Application: groups of order $pq$. Let $p < q$ be primes. Since $n_q \mid p$ and $n_q \equiv 1 \pmod q$, and $p < q$, we get $n_q = 1$: there is a normal Sylow $q$-subgroup $Q$. Also $n_p \mid q$ and $n_p \equiv 1 \pmod p$, so $n_p \in \{1, q\}$. If $p \nmid q - 1$, then $n_p = 1$, both Sylow subgroups are normal, and $G \cong C_p \times C_q \cong C_{pq}$ is cyclic. If $p \mid q - 1$, nonabelian groups of order $pq$ exist, realized as semidirect products (see §17).

Application: groups of order $2p$. Let $p$ be an odd prime. Then $n_p \mid 2$ and $n_p \equiv 1 \pmod p$, so $n_p = 1$: there is a normal $P \cong C_p$. A Sylow $2$-subgroup $H \cong C_2$ exists, and $G = P H$ with $P \cap H = \{e\}$, so $G \cong C_p \rtimes C_2$. The action is by an automorphism of order dividing $2$; since $\operatorname{Aut}(C_p) \cong C_{p-1}$ has a unique element of order $2$, namely inversion, it is trivial or inversion. Hence every group of order $2p$ is cyclic $C_{2p}$ or dihedral $D_p$ (see §16).

Part IV: Important Classes of Groups

14. Cyclic Groups

Every cyclic group is isomorphic to exactly one of the groups $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$ with $n \geq 1$; we write $C_n$ for the abstract cyclic group of order $n$.

Every subgroup of a cyclic group is cyclic. The subgroups of $C_n$ are in bijection with the divisors of $n$: for each $d \mid n$ there is exactly one subgroup of order $d$. The number of generators of $C_n$ is $\varphi(n)$, the Euler totient, these being the classes of the integers coprime to $n$. Cyclic groups are abelian, and every quotient of a cyclic group is cyclic.

15. Symmetric and Alternating Groups

The symmetric group $S_n$ is the group of all permutations of $\{1, \ldots, n\}$, of order $n!$. Every permutation is a product of pairwise disjoint cycles, uniquely up to the order of the factors; disjoint cycles commute, and the order of a permutation is the least common multiple of its cycle lengths.

A transposition is a $2$-cycle. Every permutation is a product of transpositions, and $S_n$ is generated by the adjacent transpositions $(1\,2), (2\,3), \ldots, (n-1\,n)$, or by $(1\,2)$ together with the $n$-cycle $(1\,2 \cdots n)$.

The sign $\operatorname{sgn}(\sigma) = (-1)^N$, where $N$ is the number of transpositions in any expression of $\sigma$, is well-defined, and $\operatorname{sgn} : S_n \to \{\pm 1\}$ is a surjective homomorphism onto the cyclic group of order $2$. Its kernel is the alternating group $A_n$ of even permutations, a normal subgroup of index $2$ for $n \geq 2$, so $|A_n| = n!/2$. For $n \geq 3$, $A_n$ is generated by the $3$-cycles, and $A_n$ is simple for $n \geq 5$. The group $A_4$ is not simple: its three double transpositions together with the identity form a normal subgroup isomorphic to $C_2 \times C_2$.

Two permutations are conjugate in $S_n$ if and only if they have the same cycle type, so the conjugacy classes of $S_n$ correspond to the partitions of $n$. For $n \geq 3$, the center $Z(S_n)$ is trivial.

16. Dihedral Groups

For $n \geq 3$, the dihedral group $D_n$ is the group of order $2n$ with presentation

$$ D_n = \langle r, s \mid r^n = s^2 = e, \ s r s = r^{-1} \rangle, $$

and every element is uniquely $r^i s^j$ with $0 \leq i < n$ and $j \in \{0, 1\}$. The cyclic subgroup $\langle r \rangle \cong C_n$ is normal of index $2$, so

$$ D_n \cong C_n \rtimes C_2. $$

For $n \geq 3$, $D_n$ is nonabelian. Its center is $\{e, r^{n/2}\}$ for even $n$ and is trivial for odd $n$. The smallest genuine case is $D_3 \cong S_3$. Some authors write $D_{2n}$ for the group of order $2n$; here the subscript records the number of sides.

17. Direct and Semidirect Products

The direct product $G \times H$ is the set of pairs $(g, h)$ with componentwise operation $(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)$, a group of order $|G| \, |H|$. Internally, $G$ is the direct product of subgroups $H, K$ if

$$ G = H K, \qquad H \cap K = \{e\}, \qquad H, K \trianglelefteq G. $$

Examples are the Klein four group $V_4 = C_2 \times C_2$, and the Chinese remainder isomorphism $\mathbb{Z}/mn\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ when $\gcd(m, n) = 1$, in particular $\mathbb{Z}/6\mathbb{Z} \cong C_2 \times C_3$.

The direct product cannot produce a nonabelian group from abelian factors. The semidirect product repairs this: given groups $N$, $H$ and a homomorphism $\varphi : H \to \operatorname{Aut}(N)$, the group $N \rtimes_\varphi H$ has underlying set $N \times H$ and operation

$$ (n_1, h_1)(n_2, h_2) = (n_1 \, \varphi(h_1)(n_2), \, h_1 h_2). $$

Here $N$ is normal and $H$ is a complement. Internally, $G$ is such a product if

$$ G = N H, \qquad N \cap H = \{e\}, \qquad N \trianglelefteq G, $$

with $H$ not required to be normal. Examples: $D_n \cong C_n \rtimes C_2$; $S_n \cong A_n \rtimes C_2$ for $n \geq 2$; and, when $p \mid q - 1$ are primes, the nonabelian group of order $pq$ is $C_q \rtimes C_p$, where $C_p$ acts through an automorphism of order $p$.

18. Finitely Generated Abelian Groups

Structure theorem. Every finitely generated abelian group $G$ is isomorphic to a finite direct sum

$$ G \cong \mathbb{Z}^r \oplus \mathbb{Z}/d_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/d_k\mathbb{Z}, $$

where $r \geq 0$ and $d_1 \mid d_2 \mid \cdots \mid d_k$ with each $d_i > 1$. The integer $r$ is the rank and the $d_i$ are the invariant factors; both are uniquely determined by $G$. The group is finite if and only if $r = 0$.

Equivalently, $G$ is a direct sum of cyclic groups of prime-power order together with a free part $\mathbb{Z}^r$; the prime powers are the elementary divisors, unique up to order. A finitely generated torsion-free abelian group is free abelian.

Example. $\mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/12\mathbb{Z}$, with invariant factors $2, 12$ and elementary divisors $2, 3, 4$.

19. Free Groups and Presentations

The free group $F(X)$ on a set $X$ consists of the reduced words in the letters $x, x^{-1}$ ($x \in X$) under concatenation with cancellation; its defining property is that every function $f : X \to G$ into a group extends uniquely to a homomorphism $F(X) \to G$. Free groups of rank at least $2$ are nonabelian and torsion-free.

A presentation $\langle X \mid R \rangle$ exhibits a group as a quotient $F(X)/N$, where $N$ is the normal closure of the relations $R$; every group is a quotient of a free group, and examples include $\langle x \mid x^n \rangle$ for $C_n$ and $\langle r, s \mid r^n, s^2, (s r)^2 \rangle$ for $D_n$.

20. Matrix Groups and the Quaternion Group

For a field $F$, the center of $GL_n(F)$ is the group of scalar matrices $\{a I : a \in F^\times\} \cong F^\times$, and the determinant is a surjective homomorphism onto $F^\times$ with kernel $SL_n(F)$, so that

$$ GL_n(F) / SL_n(F) \cong F^\times. $$

Over the finite field $\mathbb{F}_q$, the general linear group is finite of order

$$ |GL_n(\mathbb{F}_q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1}) = q^{n(n-1)/2} \prod_{i=1}^{n} (q^i - 1), $$

by choosing the columns successively as linearly independent vectors. For instance $|GL_2(\mathbb{F}_2)| = 6$ and $|GL_3(\mathbb{F}_2)| = 168$.

The quaternion group is

$$ Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}, $$

with the relations $i^2 = j^2 = k^2 = i j k = -1$. It has order $8$ and is nonabelian, with element orders $1, 2, 4, 4, 4, 4, 4, 4$. Its center is $\{\pm 1\}$, and $Q_8 / Z(Q_8) \cong C_2 \times C_2$. Every subgroup of $Q_8$ is normal, the proper nontrivial ones being $\{\pm 1\}$ and the three cyclic subgroups $\langle i \rangle$, $\langle j \rangle$, $\langle k \rangle$ of order $4$. A nonabelian group all of whose subgroups are normal is called Hamiltonian; $Q_8$ is the smallest nonabelian example.

Part V: Summary

21. Summary of Group Classes

Group	Order	Abelian	Cyclic	Simple
$C_n \cong \mathbb{Z}/n\mathbb{Z}$	$n$	yes	yes	only if $n$ prime
$V_4 \cong C_2 \times C_2$	$4$	yes	no	no
$S_n$, $n \geq 3$	$n!$	no	no	no
$A_n$, $n \geq 5$	$n!/2$	no	no	yes
$D_n$, $n \geq 3$	$2n$	no	no	no
$Q_8$	$8$	no	no	no
$GL_n(F)$, $n \geq 2$	depends on $F$	no	no	no

The cyclic groups of prime order are the abelian simple groups. The Sylow theorems describe the $p$-subgroups of a finite group and constrain its possible orders, while the structure theorem for finitely generated abelian groups classifies the abelian case.

22. Groups in the Wider Corpus

The group of units $R^\times$ of a ring is a group under multiplication, recovering $(\mathbb{Z}/n\mathbb{Z})^\times$ of order $\varphi(n)$, the multiplicative groups $F^\times$ of fields, and $M_n(F)^\times = GL_n(F)$.

Summary

A group is a set with a single associative binary operation, an identity element and inverses. The elementary properties follow at once — the identity and inverses are unique and powers obey the usual laws — and subgroups, homomorphisms, cosets and Lagrange's theorem are developed from there. The normal subgroups are exactly those for which the quotient is a group, and the isomorphism theorems relate the homomorphic images to the quotients by the kernels.

The article then develops the structural theory. The center, the commutator subgroup and the abelianization measure how far a group is from abelian. Group actions, with their orbits and stabilizers, give the orbit–stabilizer theorem; the conjugation action gives the conjugacy classes and the class equation, from which the theory of $p$-groups follows. The Sylow theorems describe the subgroups of prime-power order of a finite group, and the cyclic, symmetric and alternating, and dihedral groups are the worked families.

The article closes with the constructions and the classification results: direct and semidirect products, the structure theorem for finitely generated abelian groups, free groups and presentations, and the matrix groups together with the quaternion group $Q_8$. The section on group classes gathers the standard examples in a table, and the final section places the group of units of a ring, and the groups of the wider corpus, in this setting.