3 Group Theory

The concept of a group is central to abstract algebra for its ubiquity throughout mathematics and the physical sciences: wherever one finds symmetry, one will find groups. Our primary reference for this brief review is Dummit and Foote (2003). For alternate perspectives, consider Shahriar (2017) and Artin’s well-known expositions of group theory Artin (2011). As always, the best book is the third one you read!

Many of the groups we study come from an existing additive or multiplicative context, or perhaps via abstract symmetries (that is, self-bijections preserving a desired property) of some object. Indeed, this is the origin of group theory. When studying an abstract group $G$ with some unspecified group operation $\ast$, aka the group law of $G$, we will often use a multiplicative notation: given $g, h \in G$, we write $gh$ as a shorthand for $g \ast h$, $g^{-1}$ for the inverse of $g$ with respect to $\ast$, and $g^n$ to mean \[ \overbrace{g \ast \cdots \ast g}^{n \text{ times}} \text{ if } n > 0, \] or \[ \overbrace{g^{-1} \ast \cdots \ast g^{-1}}^{-n \text{ times}}\ \text{ if } n < 0. \] If we have some fixed group where an additive notation is more appropriate—usually some abelian group—then we might instead write $g+h, -g,$ and $ng$, respectively, for these shorthands.

3.1 Notable Families

Example 3.1 (Additive groups) (Dummit and Foote 2003, 8) The sets $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C},$ and $\operatorname{Mat}_n(\mathbb{R})$ are abelian groups with respect to addition. Furthermore, the set of integers modulo $n$—the set of equivalence classes of $\mathbb{Z}$ under the relation $a \sim b$ whenever $n|(b-a)$, which is usually written $\mathbb{Z}/{n}\mathbb{Z}$—is a group with respect to addition. Colloquially, this is the clock with $n$ tickmarks.

Example 3.2 (Group of units) Let $M$ be a set equipped with a unital, associative binary operation $\ast.$ Then the subset of invertible elements, \[ M^\times \coloneqq \{ x \in M \mid x \text{ is invertible} \}, \] is a group with respect to $\ast$ known as the group of units. In particular, we write \[ \begin{aligned} \mathbb{Z}^\times &= \{1, -1\}, \\ \mathbb{Q}^\times &= \mathbb{Q}\setminus \{0\}, \\ \mathbb{R}^\times &= \mathbb{R}\setminus \{0\}, \\ \mathbb{C}^\times &= \mathbb{C}\setminus \{0\}. \end{aligned} \] for the groups with respect to multiplication.

Example 3.3 (Group of $n$th roots) (Bretscher 2013, 365) Let $n \in \mathbb{N}$ be fixed. The set $\mathcal{C}_{n} \coloneqq \{ z \in \mathbb{C}: z^n = 1 \}$ is the group of $n$th roots of unity with respect to multiplication. We can list these elements explicitly using Euler’s formula: \[ \mathcal{C}_{n} = \left\{ 1, e^{2 \pi i/n}, \dots, e^{2 \pi i (n-1)/n} \right\}. \]

Example 3.4 (Trivial group) The set $\{1\}$, often simply denoted $1$ or $\mathcal{C}_{1}$, is called the trivial group. Notice that $\mathcal{C}_{1}$ trivally satisfies the group axioms when we define $1 \cdot 1 = 1.$

Example 3.5 (Torus group) The unit circle, written as the set \[ \mathbb{T}\coloneqq \{ a+bi \in \mathbb{C}: a^2+b^2=1 \} \] of length $1$ complex numbers, is the circle (a.k.a., torus) group with respect to multiplication.

Example 3.6 (Group of invertible matrices) The general linear group of $n \times n$ matrices with entries in a field $F$, written $\operatorname{GL}_n(F)$, is the group of invertible elements in $\operatorname{Mat}_n(F).$ That is, \[ \begin{aligned} \operatorname{GL}_n(F) & \coloneqq \operatorname{Mat}_n(F)^\times \\ & = \{ M \in \operatorname{Mat}_n(F) \mid M \text{ is invertible} \} \\ & = \{ M \in \operatorname{Mat}_n(F) \mid \det(M) \not = 0 \} \end{aligned} \] is a group with respect to matrix multiplication. In general, if $V$ is an $F$-vector space, then \[ \operatorname{GL}(V) \coloneqq \{ A: V \to V \mid A \text{ is a linear isomorphism} \} \] is the group of linear automorphisms. Note that $\operatorname{GL}_n(F)$ is a non-abelian for all $n>1$; otherwise, $\operatorname{GL}_1(F) = F^\times.$ Of additional interest is \[ \operatorname{U}(n) \coloneqq \left\{ Q \in \operatorname{GL}_n(\mathbb{C}) \mid Q^{-1} = \overline{Q^\top} \right\}, \] the $n$-dimensional unitary group. We emphasize that $\operatorname{U}(1) = \mathbb{T}$, the torus group.

Example 3.7 (Permutation group) (Dummit and Foote 2003, 29) If $X$ is a set, then $\operatorname{Perm}(X)$ is used to denote the group of permutations (i.e., all self-bijections on $X$) under function composition. If $X = \{1,2,\dots,n\}$ for some $n \in \mathbb{Z}^+$, we write $\mathcal{S}_{n}$ and call this group the symmetric group on $n$ letters.

To describe permutations, we use cycle notation. For instance, \[ (1\ 2\ 4)(3\ 6) \in \mathcal{S}_{6} \] denotes the permutation with $1 \mapsto 2$, $2 \mapsto 4$, $4 \mapsto 1$, $3 \mapsto 6$, $6 \mapsto 3$, and $5$ fixed.

Example 3.8 (Quaternion group) (Dummit and Foote 2003, 36) The set \[ \mathcal{Q}_{8}\coloneqq \{1, -1, i, -i, j, -j, k, -k\} \subset \mathbb{H} \] can be made into a group by defining \[ k = ij\ \text{ and } {-1} = i^2 = j^2 = k^2 = ijk. \] Here we formally understand $-1$ as central (that is, an element commuting with all others) and an element squaring to $1$. Note that $i^{-1} = -i$, and similarly $j^{-1} = -j$ and $k^{-1} = -k$. In particular, the quaternion group is non-abelian, since we must have $ij = -ji$: \[ \begin{aligned} -1 = k^2 = (ij)(ij) & \Rightarrow iji = -j = j^{-1} \\ & \Rightarrow ij = (iji) i^{-1} = j^{-1} i^{-1} = ji. \end{aligned} \]

Example 3.9 If $G$ and $H$ are groups, with operations $\ast$ and $\star$, respectively, then the set $G \times H$ can be equipped with a group operation $\odot$ as follows: \[ (g_1,h_1) \odot (g_2,h_2) \coloneqq (g_1 \ast g_2, h_1 \star h_2). \] Often we suppress all the group operations and simply write \[ (g_1,h_1)(g_2,h_2) = (g_1 g_2, h_1 h_2). \]

Example 3.10 The group $\mathcal{V}\coloneqq \mathbb{Z}/{2}\mathbb{Z} \times \mathbb{Z}/{2}\mathbb{Z}$ is called the Klein 4-group.

3.2 Fundamentals

Definition 3.1 (Dummit and Foote 2003, 20) Let $G$ be a group with $g \in G.$ The order of $g$, written $|g|$, is the smallest $n \in \mathbb{Z}^+$ such that $g^n = e$; if no such $n$ exists, we say $g$ is of infinite order.

Example 3.11 The element $1 \in \mathbb{Z}$ has infinite order, but $1 \in \mathbb{Z}/{n}\mathbb{Z}$ has order $n.$ Remember, the group operation here is addition!

Example 3.12 The element $i \in \mathcal{C}_{4}$ has order 4. The element $e^i \in \mathbb{T}$ has infinite order.

Subgroups

Definition 3.2 (Dummit and Foote 2003, 46) Let $G$ be a group with $H \subseteq G.$ We say that $H$ is a subgroup of $G$, written $H \leq G$, if $H$ is also a group with respect to the operation of $G.$ If we want to emphasize that $H$ is a proper subgroup, i.e., $H \not = G$, then we write $H < G.$

Example 3.13 The set of multiples $n\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ for all $n \in \mathbb{N}.$

Example 3.14 Any group $G$ contains the trivial subgroup $\{e\} \leq G.$

Example 3.15 The set $\{ \mathbb{I}, (1 \ 2 \ 3), (1 \ 3 \ 2) \}$ is a subgroup of $\mathcal{S}_{3}.$

Example 3.16 There is a chain of subgroups $\mathbb{Z}< \mathbb{Q}< \mathbb{R}< \mathbb{C}$ with respect to addition. Similarly, there is a chain $\{\pm 1\} = \mathbb{Z}^\times < \mathbb{Q}^\times < \mathbb{R}^\times < \mathbb{C}^\times$ with respect to multiplication.

Example 3.17 For any group $G$ and fixed $g \in G$, the set \[ \langle g \rangle \coloneqq \{ g^n \mid n \in \mathbb{Z}\} \] is a subgroup of $G$ of order $|g|.$

Theorem 3.1 (Dummit and Foote 2003, 62) If $H, K \leq G$, then $H \cap K \leq G.$ More generally, if $\{H_\lambda\}_{\lambda \in \Lambda}$ is an arbitrary collection of subgroups in $G$, then $\bigcap_{\lambda \in \Lambda} H_\lambda \leq G$

Theorem 3.2 (Lagrange) (Dummit and Foote 2003, 89) Let $G$ be a finite group. If $H \leq G$, then $|H|$ divides $|G|.$

Corollary 3.1 If $G$ is finite and $g \in G$, then $|g|$ divides $|G|.$ In particular, $g^{|G|} = e.$

Corollary 3.2 (Fermat’s little theorem) Fix a prime $p.$ Then $a^p = a$ for all $a \in \mathbb{Z}/{p}\mathbb{Z}.$

Conjugacy

Theorem 3.3 (Dummit and Foote 2003, 50) Let $G$ be a group. The center of $G$, written $\mathbf{Z}(G)$, is \[ \mathbf{Z}(G) \coloneqq \{ h \in G \mid gh = hg \text{ for all } g \in G \} \leq G. \] That is, $\mathbf{Z}(G)$ consists of the elements that commute with every element in $G.$

Example 3.18 For all $n > 2$, we have $\mathbf{Z}(\mathcal{S}_{n}) = \{\mathbb{I}\}.$

Definition 3.3 (Dummit and Foote 2003, 123) If $G$ is a group with $g,h \in G$, we say that $g h g^{-1}$ is the conjugate of $h$ by $g.$ Conjugacy defines an equivalence relation on $G$: \[ g \sim g' \text{ means } g' = h g h^{-1} \text{ for some } h \in G. \] The equivalence classes of this relation are called conjugacy classes and are denoted \[ \operatorname{cl}_{G}({g}) \coloneqq \{ h g h^{-1} \mid h \in G \}, \] or simply $\operatorname{cl}_{}({g})$ when the group is clear from context.

Example 3.19 If $G$ is an abelian group, then each element is only conjugate to itself; more generally, for every element $g \in \mathbf{Z}(G)$ we have $\operatorname{cl}_{G}({g}) = \{g\}.$

Example 3.20 In light of the formula \[ \sigma (1 \ 2 \ \cdots \ k ) \sigma^{-1} = (\sigma(1) \ \sigma(2) \ \cdots \ \sigma(k) ), \] the conjugacy classes of $\mathcal{S}_{n}$ are in bijective correspondence with partitions of $n.$ Ino ther words, the conjugacy class of a permutation is given by its cycle type.

Example 3.21 In the group $\operatorname{GL}_n(F)$, conjugation is simply change of basis (this is usually written slightly differently, cf. Theorem 2.8, where we replace a matrix $A$ with $S^{-1} A S$ and the columns of $S$ represent the new basis). In particular, diagonalizable matrices are those that are conjugate to a diagonal matrix.

Proposition 3.1 For any group $G$, any conjugate of $g \in G$ has the same order as $g$ itself.

Generators and Relations

Definition 3.4 (Dummit and Foote 2003, 26) Let $G$ be a group. A (possibly infinite) collection of elements $S \subset G$ is said to generate $G$, written $G = \langle S \rangle$, if every element of $G$ can be written as a finite product of elements in $S$ and their inverses. Alternatively but equivalently, define \[ \langle S \rangle \coloneqq \bigcap_{\substack{H \leq G \\ S \subseteq H}} H \] as the subgroup generated by $S.$ From this definition, it is evident by inspection that $\langle S \rangle$ is the smallest subgroup of $G$ containing $S$. If $G$ can be generated by a single element, $G = \langle g \rangle$, then we say that $G$ is cyclic.

Example 3.22 The set $\{1\}$ generates the integers $\mathbb{Z}$ (for this reason, the group $\mathbb{Z}$ is often called the infinite cyclic group}. Generating sets are not unique; for example, the sets $\{-1\}$ and $\{2,9\}$ also generate $\mathbb{Z}.$ Similarly, $\{1\}$ generates $\mathbb{Z}/{n}\mathbb{Z}.$ On the other hand, $e^{2 \pi i/n}$ generates $\mathcal{C}_{n}.$

Example 3.23 The elements $\{i,j\}$ generate the quaternion group $\mathcal{Q}_{8}.$

Example 3.24 The Klein 4-group $\mathcal{V}$ is the smallest non-cyclic group.

Definition 3.5 (Dummit and Foote 2003, 218) Given a generating set $S$ of a group $G$, equations satisfied by the generators are called relations. If $R_1, \dots, R_n$ is some finite list of relations from which the group law $G$ can be completely determined, along with the standing assumptions of associativity and identity and inverses, we call the generators and relations a presentation for $G$ and write \[ G = \langle S \mid R_1, \dots, R_m \rangle. \]

Example 3.25 We can present $\mathcal{C}_{n}$ as $\langle \zeta \ | \ \zeta^n=1 \rangle.$ We might imagine $\zeta = e^{2 \pi i/n}$—though this identification is not necessary, and potentially misleading!

Example 3.26 A presentation for the quaternion group is \[ \mathcal{Q}_{8}= \langle i,j \ | \ i^4=1, i^2=j^2, jij = i \rangle. \] Note that we make no mention of $k \in \mathcal{Q}_{8}$; it is simply understood as the product $ij.$ More generally, the quaternion group of order $4n$, denoted $\mathcal{Q}_{4n}$, has the presentation \[ \mathcal{Q}_{4n} = \langle a,j \ | \ a^{2n}=1, a^n=j^2, j^{-1} a j = a^{-1} \rangle. \] In terms of the quaternion algebra $\mathbb{H}$, we can think of $a = e^{\pi i/n}$.

Example 3.27 The Klein 4-group $\mathcal{V}$ can be presented as $\langle a, b \ | \ a^2 = b^2 = (ab)^2 = e \rangle.$

Example 3.28 (Dummit and Foote 2003, 23) The dihedral group $\mathcal{D}_{2n}$, alternatively understood as the group of $2n$ rigid symmetries for a regular $n$-gon,¹ can be presented as \[ \langle r, s \ | \ r^n = \mathbb{I}, s^2 = \mathbb{I}, rs = sr^{-1} \rangle. \]

When thinking of $\mathcal{D}_{2n}$ as acting on a regular $n$-gon with its vertices labeled $1, \dots, n$ in a clockwise fashion, we will use the following convention:

$r$ is a rotation clockwise by $\tfrac{2\pi}{n}$
$s$ is a flip across the axis between the center of the $n$-gon and the vertex labeled $1.$

One can show that \[ \mathbf{Z}(\mathcal{D}_{2n}) = \left\{ \begin{array}{ll} \{e\} & n \text{ odd} \\ \{e,r^{n/2}\} & n \text{ even.} \\ \end{array} \right. \]

3.3 Homomorphisms

Basic Properties

Definition 3.6 (Dummit and Foote 2003, 36) Let $G$ and $H$ be groups with the operations $\ast$ and $\star$, respectively. A function $\rho: G \to H$ is called a (group) homomorphism if \[ \rho(g \ast g') = \rho(g) \star \rho(g') \text{ for all } g, g' \in G. \] If $\rho$ is also a bijection², then $\rho$ is called an isomorphism and we write $G \cong H$ to denote the existence of such a $\rho$ (pronounced “$G$ is isomorphic to $H.$”)

Proposition 3.2 If $\rho: G \to H$ is a homomorphism, then $\rho(e_G) = e_H$ and $\rho(g^{-1}) = \rho(g)^{-1}$ for all $g \in G.$ Moreover, if $\rho: G \to H$ is a bijection, then $\rho^{-1}: H \to G$ is also a homomorphism.

Proposition 3.3 If $\rho: G \to K$ and $\sigma: K \to H$ are homomorphisms, then $\sigma \circ \rho: G \to H$ is also a homomorphism. If $\rho$ and $\sigma$ are isomorphisms, then so is their composition.

Example 3.29 For any groups $G$ and $H$, there is always a homomorphism $\rho: G \to H$ given by $\rho(g) = e_H$ for all $g \in G.$ This is called the trivial homomorphism.

Example 3.30 The groups $\mathcal{D}_{6}$ and $\mathcal{S}_{3}$ are isomorphic. One can write down an isomorphism between them by considering the action of $\mathcal{D}_{6}$ on the vertices of an equilateral triangle as in Example 3.28. The induced homomorphism is given by \[ \begin{aligned} e & \mapsto \mathbb{I}, \\ r & \mapsto (1\ 2\ 3), \\ r^2 & \mapsto (1\ 3\ 2), \\ s & \mapsto (2\ 3), \\ sr & \mapsto (1\ 3), \\ sr^2 & \mapsto (1\ 2). \end{aligned} \]

Theorem 3.4 If $\rho: G \to H$ is a homomorphism and $g \in G$ has $|g|<\infty$, then $|\rho(g)|$ divides $|g|.$

Corollary 3.3 If $\rho: G \to H$ is an isomorphism, then $|g|=|\rho(g)|$ for all $g \in G.$

Example 3.31 There is a homomorphism $\rho: \mathbb{T}\to \mathbb{T}$ given by $z \mapsto z^2$ which, geometrically, wraps the circle around itself twice. Note that $i \in \mathbb{T}$ has order $4$ but $\rho(i)=-1$ has order $2.$

Example 3.32 If $V$ is an $n$-dimensional $F$-vector space, fixing a basis $\mathscr{B} \subset V$ gives rise to a group isomorphism $\operatorname{GL}(V) \cong \operatorname{GL}_n(F)$ by sending $L \mapsto {}_{\mathscr{B}}[L]_{\mathscr{B}}.$

Example 3.33 (Permutation representation) For any $n \in \mathbb{Z}^+$, there is a map $\mathcal{S}_{n} \to \operatorname{GL}_n(\mathbb{Z})$ given by sending a permutation $\sigma$ to the corresponding permutation matrix. For example, when $n=3$, this homomorphism assigns \[ (1\ 2) \mapsto \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \ \text{ and } \ (1\ 2\ 3) \mapsto \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}. \]

Example 3.34 (Determinants) For any field $F$ and $n \in \mathbb{Z}^+$, there is a homomorphism $\operatorname{GL}_n(F) \to F^\times = F-\{0\}$ given by the determinant, $A \mapsto \det(A).$ This is a rebranding of the important fact from linear algebra (cf. Definition 2.15): \[ \det(AB) = \det(A) \det(B). \]

More generally, if $R$ is any commutative ring, we can write $\operatorname{GL}_n(R)$ for the invertible $n \times n$ with entries in $R$ and write down a determinant homomorphism $\operatorname{GL}_n(R) \to R^\times.$

Definition 3.7 If $\rho: G \to H$ is a group homomorphism, then \[ \ker \rho \coloneqq \{ g \in G \mid \rho(g) = e_H \} \leq G \] is called the kernel of $\rho$; the image of $\rho$ is defined as: \[ \operatorname{im}\rho \coloneqq \{ h \in H: h = \rho(g) \text{ for some } g \in G \} \leq H. \]

Example 3.35 The kernel of the determinant map is called $\operatorname{SL}_n(F)$, the special linear group.

Example 3.36 (Dummit and Foote 2003, 107) There is a map $\varepsilon: \mathcal{S}_{n} \to \{ \pm 1 \}$ obtained by composing the permutation representation with the matrix determinant. For example, when $n=3$: \[ (1\ 2) \mapsto \det\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}=-1 \quad \text{ and } \quad (1\ 2\ 3) \mapsto \det\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}=1. \] The kernel of this homomorphism is called the alternating group, written $\mathcal{A}_{n} \leq \mathcal{S}_{n}.$ The elements of $\mathcal{A}_{n}$ are called even and those in $\mathcal{S}_{n} \setminus \mathcal{A}_{n}$ are called odd.

Isomorphisms

Theorem 3.5 A homomorphism $\rho: G \to H$ is one-to-one if and only if $\ker \rho = \{e_G\}.$ Moreover, one can show that if $\rho(g) = h$, then the fiber $\rho^{-1}(h)$ can be described as the set \[ g \ker \rho \coloneqq \{ g k \mid k \in \ker \rho\}. \] This is the (left) coset (see Section 3.3.3) of $g$ with respect $\ker \rho.$

This Theorem should be reminiscent of Theorem 2.5.

Remark 3.1. If $\rho: G \to H$ is a homomorphism and $B \leq G$, then $\rho(B)$ is a subgroup of $\operatorname{im}\rho$ in $H.$ That is, the image of a subgroup is a subgroup. On the other hand, if $C \leq H$, then the inverse image $\rho^{-1}(C)$ is a subgroup of $G$ containing $\ker \rho.$

Example 3.37 Consider the map $\rho: \mathcal{D}_{8} \to \mathcal{V}$ given by \[ \begin{gathered} \rho(e) = \rho(r^2) = (0,0), \\ \rho(r) = \rho(r^3) = (1,0), \\ \rho(s) = \rho(sr^2) = (0,1), \\ \rho(sr) = \rho(sr^3) = (1,1). \end{gathered} \] The subgroup $\langle r \rangle$ maps to the subgroup $\langle (1,0) \rangle$; the inverse image of $\langle (0,1) \rangle$ is $\langle s, r^2 \rangle.$

Remark 3.2. When defining a homomorphism $\rho: G \to H$, where $G = \langle S \mid R_1, \dots, R_m \rangle$, one need only define $\rho$ on the generating set $S.$ This is simply because every $g \in G$ can be built with elements in $S$, and so the homomorphism already “knows” how to evaluate on each $g.$ To check that $\rho$ is well-defined, we verify that the relations $R_i$ are preserved by $\rho.$

Example 3.38 If we try to build a homomorphism $\rho: \mathbb{Z}/{n}\mathbb{Z} \to \mathbb{Z}$ by defining $\rho(1) \coloneqq 1$, we will find that such a map is not well-defined, i.e., it cannot exist. Indeed, the relation \[ \underbrace{1 + \cdots + 1}_n = 0 \] from $\mathbb{Z}/{n}\mathbb{Z}$ becomes the (false!) statement $n=0$ over the integers. From this reasoning, we can see that the only homomorphism $\mathbb{Z}/{n}\mathbb{Z} \to \mathbb{Z}$ is the trivial homomorphism.

Example 3.39 We can build a homomorphism $\rho: \mathcal{D}_{12} \to \mathbb{Z}/{2}\mathbb{Z}$ by defining \[ \begin{gathered} \rho(r) \coloneqq 0, \\ \rho(s) \coloneqq 1. \end{gathered} \] If we wish to compute, say, $\rho(sr^3)$, we can apply the homomorphism law: \[ \rho(sr^3) = \rho(s) + \rho(r^3) = \rho(s) + 3 \rho(r) = 1 + 0 = 1. \] For well-defined-ness, we check that $r^6 = e$, $s^2 = e$, and $rs=sr^{-1}$ hold after applying $\rho$: \[ \begin{gathered} \rho(r^6) = 6 \rho(r) = 0 = \rho(e) \\ \rho(s^2) = 2 \rho(s) = 2 = 0 = \rho(e) \\ \rho(rs) = \rho(r)+\rho(s) = 0+1 = 1 = 1-0 = \rho(s) - \rho(r) = \rho(sr^{-1}). \end{gathered} \]

Example 3.40 ($\mathbb{Z}$ is free) Given a group $G$ and element $g \in G$, there is a unique homomorphism $\mathbb{Z}\to G$ given by $1 \mapsto g$ and hence $n \mapsto g^n$ for all $n.$ The image of this homomorphism is $\langle g \rangle.$ We do not need to check if this map is well-defined because there are no relations on $1 \in \mathbb{Z}$ to check.

Example 3.41 (Finite cyclic groups) The groups $\mathbb{Z}/{n}\mathbb{Z}$ and $\mathcal{C}_{n}$ are isomorphic; the former is sometimes called the additive cyclic group of order $n$ while the latter is the multiplicative cyclic group of order $n.$ We can give an explicit isomorphism by sending $1 \mapsto e^{2 \pi i/n}$ and hence $k \mapsto e^{2 \pi i k/n}$ for all $k$; note that this map is well-defined since the relation in $\mathbb{Z}/{n}\mathbb{Z}$, \[ \underbrace{1+\dots+1}_{n \text{ times}}=0, \] passes to the (true) relation $\underbrace{e^{2 \pi i/n} \cdot \dots \cdot e^{2 \pi i/n}}_{n \text{ times}}=1$ in $\mathcal{C}_{n}.$

Remark 3.3. If $\rho: G \to H$ is a one-to-one map, then $G \cong \operatorname{im}\rho.$ In such a context, we might even say that $G$ is a subgroup of $H$ by identifying $G$ with its image, by which we understand $\rho$ as an inclusion.

Example 3.42 We can identify $\mathcal{C}_{n} < \mathcal{D}_{2n}$ by the map $e^{2 \pi i/n} \mapsto r.$ Similarly, we can identify $\mathcal{D}_{2n} \leq \mathcal{S}_{n}$ by the action of $\mathcal{D}_{2n}$ on the vertices of a regular $n$-gon. We can even identify \[ \{\mathbb{I}\} = \mathcal{S}_{1} < \mathcal{S}_{2} < \mathcal{S}_{3} < \dots < \mathcal{S}_{n} < \mathcal{S}_{n+1} < \dots \]

by thinking of a permutation $\sigma \in \mathcal{S}_{n}$ as acting trivially on any number $k > n.$ Lastly, one can identify $\mathcal{V}\leq \mathcal{S}_{4}$ by using, for instance, the homomorphism $(1,0) \mapsto (1\ 2)$ and $(0,1) \mapsto (3\ 4).$

Remark 3.4 (Non-isomorphism criteria). Given two groups $G$ and $H$, one might want to prove they are not isomorphic. In this case, the following criteria can come in handy. $G$ and $H$ cannot be isomorphic if:

$G$ and $H$ have different cardinalities.
One of $G$ and $H$ is abelian but the other is not.
For some fixed $n \in \mathbb{Z}^+$, $G$ and $H$ contain a different number of elements of order $n.$
$G$ and $H$ have different subgroup lattices.

These criteria are listed by their difficulty to assess. Beware: there exist non-isomorphic groups which satisfy all these criteria and have to be discerned by more sophisticated techniques. We shall establish additional criteria throughout this course.

Definition 3.8 There is an equivalence relation on the collection (category) of all groups: \[ G \sim H \text{ means that there is an isomorphism } G \xrightarrow{\cong}H. \] The equivalence classes of this relation are called the isomorphism types. A foundational problem in group theory is to distinguish groups, i.e., to determine whether or not a given pair of groups belong to the same isomorphism type. For small $|G| = n$, we can write out representatives of these isomorphism types as follows:

$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$\mathcal{C}_{1}$	$\mathcal{C}_{2}$	$\mathcal{C}_{3}$	$\mathcal{C}_{4}$	$\mathcal{C}_{5}$	$\mathcal{C}_{6}$	$\mathcal{C}_{7}$	$\mathcal{C}_{8}$	$\mathcal{C}_{9}$	$\mathcal{C}_{10}$	$\mathcal{C}_{11}$	$\mathcal{C}_{12}$
			$\mathcal{V}$		$\mathcal{S}_{3}$		$\mathcal{C}_{4} \times \mathcal{C}_{2}$	$\mathcal{C}_{3} \times \mathcal{C}_{3}$	$\mathcal{D}_{10}$		$\mathcal{C}_{6} \times \mathcal{C}_{2}$
							$\mathcal{V}\times \mathcal{C}_{2}$				$\mathcal{A}_{4}$
							$\mathcal{D}_{8}$				$\mathcal{D}_{12}$
							$\mathcal{Q}_{8}$				$\mathcal{Q}_{12}$

Given a group $G$ of order 6, one could use this table to see that $G$ is isomorphic to either $\mathcal{C}_{6}$ or $\mathcal{S}_{3}$ (which are not isomorphic to one another). An easy way to tell which of these $G$ shares isomorphism types with is by checking whether or not $G$ is abelian, or whether $G$ contains an element of order $6.$

Cosets and Quotients

Definition 3.9 (Dummit and Foote 2003, 77) Let $G$ be a group with $H \leq G$ and $g \in G.$ We define \[ gH \coloneqq \{ gh \mid h \in H \} \quad \text{ and } \quad Hg \coloneqq \{ hg \mid h \in H \} \] to be the left and right cosets, respectively, of $g$ with respect to $H.$ We write $G/H$ for the set of left cosets. The index³ of $H$ in $G$, denoted $\left[G:H\right]$, is the cardinality $|G/H|.$ The Lagrange theorem (Dummit and Foote 2003, 89) says that \[ |G| = \left[G:H\right]\ |H| \] and its proof relies on the fact that cosets are all the same size (cf. Theorem 3.5). We also will often refer to the conjugation of $H$ by $g$: \[ gHg^{-1} \coloneqq \{ ghg^{-1} \mid h \in H \}. \] Note that, in general, cosets are not subgroups (if $g \not \in H$, then $gH$ cannot contain the identify element). However, the conjugations of $H$ are still subgroups of $G.$

Example 3.43 Consider $H = \langle s \rangle$ in $\mathcal{D}_{8}.$ We can list some left and right cosets: \[ \begin{gathered} eH = \{ e, s \} = He = sH = Hs, \\ rH = \{ r, sr^3 \}, \\ Hr = \{ r, sr \}. \end{gathered} \] We can also consider $N = \langle r^2 \rangle$, for which we have \[ \begin{gathered} eN = \{ e, r^2 \} = Ne = r^2H = Nr^2, \\ rN = \{ r, r^3 \} = Nr = r^3N = Nr^3, \\ sN = \{ s, sr^2 \} = Ns = sr^2 N = N sr^2, \\ sr N = \{ sr, sr^3 \} = N sr = sr^3 N = N sr^3. \end{gathered} \]

Proposition 3.4 If $H \leq G$ and $g \in G$, then $gH = H$ if and only if $g \in H.$

Definition 3.10 (Dummit and Foote 2003, 82) Let $G$ be a group with $N \leq G.$ We say that $N$ is normal in $G$ and write $N \unlhd G$ if $gN = Ng$ for all $g \in G$, i.e., all the left cosets of $N$ equal the right cosets.

Remark 3.5. (Dummit and Foote 2003, 82) There are many equivalent notions of normality. For example, a subgroup $N$ is normal if and only if $gNg^{-1} = N$ for all $g \in G.$ Often the formulation that is easiest to check is $N \unlhd G$ if and only if $g x g^{-1} \in N$ for every $x \in N, g \in G.$ Moreover, $N \leq G$ is normal if and only if $N$ is the kernel of some homomorphism $G \to H.$ Lastly, a subgroup $N \leq G$ is normal if and only if $N$ can be written as a union of conjugacy classes of $G$.

Example 3.44 In Example 3.43, we saw that $rH \not = Hr$ and hence the subgroup $H = \langle s \rangle$ cannot be normal in $\mathcal{D}_{8}.$ However, $N = \langle r^2 \rangle$ is a normal subgroup of $\mathcal{D}_{8}$!

Proposition 3.5 If $\rho: G \to H$ is a homomorphism, then $\ker \rho \unlhd G.$

Definition 3.11 (Quotient groups) If $N \unlhd G$, then the set of cosets $G/N$ is a group with the operation \[ (gN)(hN) \coloneqq (gh) N. \] We call $G/N$ the quotient group of $N$ by $G.$ In words, $G/N$ is what happens if we collapse the subgroup $N$ in $G$ to a single element.

Remark 3.6. Because of how the group operation in the quotient is defined, when manipulating elements we often deal in terms of representatives (any $g' \in gN$ is a representative for the coset $gN$). Thus, we should always be careful that our calculations are well-defined.

Remark 3.7. In the quotient $G/N$, every $x \in N$ represents the identity coset.

Example 3.45 Every $G$ contains at least the normal subgroups $G \unlhd G$ and $\{e_G\} \unlhd G.$ Notice that $G/G = \{ eG \}$ is isomorphic to the trivial group $\mathcal{C}_{1}$, since we have collapsed all of $G$ to a single point. On the other hand, $G/\{e_G\}$ is isomorphic to $G$ because we did not do any collapsing at all.

Theorem 3.6 (Noether’s isomorphism theorem) (Dummit and Foote 2003, 97) If $\rho: G \to H$ is a group homomorphism and $K = \ker \rho$, there is an induced isomorphism $G/K \cong \operatorname{im}\rho$ along with inclusion-respecting bijections: \[ \{ \text{subgroups of $G$ containing $K$} \} \leftrightarrow \{ \text{subgroups of $G/K$} \} \leftrightarrow \{ \text{subgroups of $\operatorname{im}\rho$} \}. \]

Example 3.46 The homomorphism $\rho: \mathbb{R}\to \mathbb{T}$ given by $\rho(t) = e^{2\pi i t}$ has $\ker \rho = \mathbb{Z}.$ The quotient $\mathbb{R}/\mathbb{Z}$ is called the real line modulo $1$ and behaves algebraically like a circle: notice that \[ \left(\tfrac{1}{2} + \mathbb{Z}\right) + \left(\tfrac{1}{2} + \mathbb{Z}\right) = 1 + \mathbb{Z}= 0 + \mathbb{Z}. \] This reflects the induced isomorphism $\mathbb{R}/\mathbb{Z}\cong \mathbb{T}$ given by $t + \mathbb{Z}\mapsto e^{2\pi i t}.$

Example 3.47 In Example 3.44, we saw that $N = \langle r^2 \rangle \unlhd\mathcal{D}_{8}.$ The quotient \[ \mathcal{D}_{8}/N = \{ eN, rN, sN, sr N \} \] is a group since $N$ is normal. Moreover, $\mathcal{D}_{8}/N$ is isomorphic to the Klein 4-group via $\rho: \mathcal{D}_{8}/N \to \mathcal{V}$ defined by \[ \begin{gathered} \rho(eN) = (0,0), \\ \rho(rN) = (1,0), \\ \rho(sN) = (0,1), \\ \rho(srN) = (1,1). \end{gathered} \]

Example 3.48 (Dummit and Foote 2003, 82) Given $N \unlhd G$, one may consider the canonical homomorphism $G \to G/N$ given by $g \mapsto gN.$ This homomorphism is surjective and has kernel $N.$ In particular, a subgroup of $G$ is normal if and only if it is the kernel of some homomorphism.

Remark 3.8 (Sub/quotient-object duality). Throughout algebra, one-to-one maps $X \to Y$ correspond to subobjects of $Y$; surjective maps $X \to Y$ correspond to quotients of $X.$

3.4 Group Actions

Remark 3.9. Since their official inception in early 19th century work on solving equations, groups were made to act—the original definition of group was in terms of the permutations of roots of polynomials that preserve algebraic relations over $\mathbb{Q}$ (i.e., subgroups of $\mathcal{S}_{n}$). This is reconciled with our modern definition via the Cayley theorem (Dummit and Foote 2003, 120).

Basic Properties

Definition 3.12 (Dummit and Foote 2003, 41) Let $G$ be a group and $X$ a set. A group action, sometimes denoted by the shorthand $G ⟳X$, is a homomorphism $\rho: G \to \operatorname{Perm}(X).$ We often suppress $\rho$ and simply write $g \cdot x$ to mean \[ g \cdot x \coloneqq \rho(g)(x) \] Many books refer to a set $X$ equipped with such an action a $G$-set.

Example 3.49 For any group $G$ and set $X$, there is always the trivial action given by the trivial homomorphism $G \to \operatorname{Perm}(X)$, i.e., $g \cdot x = x$ for all $g \in G$ and $x \in X.$

Example 3.50 (Dummit and Foote 2003, 92) The group $\mathcal{A}_{4}$ acts on a regular tetrahedron by rotation: given a labeling of the vertices, there is exactly one even permutation corresponding to the action of each rigid symmetry on those vertices. Moreover, this correspondence sends compositions of rotations to composition of permutations.

Example 3.51 (Left multiplication) A group $G$ always acts on itself by left multiplication: \[ \begin{aligned} G & \to \operatorname{Perm}(G) \\ g & \mapsto (x \mapsto gx). \end{aligned} \]

Example 3.52 (Conjugation) A group $G$ always acts on itself by conjugation: \[ \begin{aligned} G & \to \operatorname{Perm}(G) \\ g & \mapsto (x \mapsto gxg^{-1}). \end{aligned} \]

Example 3.53 (Left cosets) Given any $H \leq G$, the group $G$ acts on $G/H$ by left multiplication: \[ \begin{aligned} G & \to \operatorname{Perm}(G/H) \\ g & \mapsto (xH \mapsto (gx)H). \end{aligned} \]

Orbit–Stabilizer Theorem

Definition 3.13 If $G ⟳X$ and $x \in X$, then the orbit $\operatorname{Orb}_{G}({x})$ of $x$ is the set of all elements $y \in Y$ that can be reached from $x$ via the $G$-action. More precisely, \[ \operatorname{Orb}_{G}({x}) \coloneqq \{ y \in X \mid y = g \cdot x \text{ for some } g \in G \}. \] Orbits are equivalence classes for the relation \[ x \sim y \text{ means } y = g\cdot x \text{ for some } g \in G. \] In particular, the orbits partition $X.$

Example 3.54 Consider the circle group $\mathbb{T}$ acting on $\mathbb{C}$ by multiplication. The orbits of this action can be visualized as circles centered at the origin of any positive radius $r$, along with the origin itself as a singleton orbit.

Definition 3.14 If $G ⟳X$ via $\rho: G \to \operatorname{Perm}(X)$, then the kernel of the action is the subgroup of elements in $G$ that act trivially on $X.$ In symbols: \[ \ker(G ⟳X) \coloneqq \ker(\rho) = \{ g \in G \mid g \cdot x = x \text{ for all } x \in X \} \unlhd G. \] For a fixed $y \in X$, we can identify the elements that leave specifically $y$ unchanged: \[ \operatorname{Stab}_{G}({y}) \coloneqq \{ g \in G \mid g \cdot y = y \}. \] This subgroup is the stabilizer of $y.$⁴ Note that $\ker(G ⟳X) \leq \operatorname{Stab}_{G}({y}) \leq G$ and \[ \ker(G ⟳X) = \bigcap_{x \in X} \operatorname{Stab}_{G}({x}). \] In addition, the fixed set of $G ⟳X$ is given by \[ X^G \coloneqq \{ x \in X \mid g \cdot x = x \text{ for all } g \in G \}. \]

Remark 3.10. Often when considering a group action $G ⟳X$, one might restrict to a subgroup $H \leq G$ by considering only how the elements of $H$ act on $X.$ That is, we have an associated action $H ⟳X.$ In this case, for an element $x \in X$, we have $\operatorname{Stab}_{H}({x}) = \operatorname{Stab}_{G}({x}) \cap H.$

Remark 3.11. Another common notation for the fixed set of $G ⟳X$ is $\operatorname{Fix}(G) \coloneqq X^G$; more generally, if $H \leq G$, we write \[ \operatorname{Fix}(H) = X^H \coloneqq \{ x \in X \mid h \cdot x = x \text{ for all } h \in H \}, \] and, for a specific element $g \in G$, we write simply $\operatorname{Fix}(g) \coloneqq \operatorname{Fix}(\langle g \rangle)$.

Example 3.55 When $G ⟳X$ trivially, the kernel of the action is all of $G.$

Example 3.56 Consider the action of $\mathcal{D}_{8}$ on the two diagonals of a square, so that $r$ and $sr$ both act by interchanging the diagonals (recall our convention in Example 3.28). We can see that the kernel of this action is $K = \langle r^2, s \rangle$, since rotation by $180^\circ$ and flipping across a diagonal axis do not interchange the diagonals.

Example 3.57 Consider the natural action of $\mathcal{S}_{n}$ on the set $X = \{1,\dots,n\}$, which has trivial kernel. For any $x \in X$, we can see that \[ \operatorname{Stab}_{\mathcal{S}_{n}}({x}) = \{ \sigma \in \mathcal{S}_{n} \mid \sigma(x) = x \} \cong \mathcal{S}_{n-1}. \]

Example 3.58 When $G ⟳G$ by left multiplication, the stabilizer of every $x \in G$ is trivial in light of the cancellation law: $gx = gy$ always means $x = y.$ Moreover, the orbit of any $x \in G$ is all of $G$, since $y = (yx^{-1}) \cdot x$ for any $y \in G.$ Actions with only one orbit are called transitive.

Example 3.59 When $G ⟳G$ by conjugation, the stabilizer of $x \in G$ is called the centralizer: \[ \operatorname{C}_{G}({x}) \coloneqq \{ g \in G \mid g x g^{-1} = x \}. \] Remember that the kernel is equal to the intersection of all stabilizers, i.e., \[ \mathbf{Z}(G) = \bigcap_{g \in G} \operatorname{C}_{G}({g}). \] Lastly, the orbit of $x$ is the conjugacy class $\operatorname{cl}_{G}({x}).$

Theorem 3.7 Suppose $G ⟳X$ and fix $g \in G$ and $x \in X.$ Then \[ \operatorname{Stab}_{G}({g \cdot x}) = g \operatorname{Stab}_{G}({x}) g^{-1}. \]

Example 3.60 When $G ⟳G/H$ by left multiplication, the stabilizer of the identity coset $eH$ is simply $H.$ This action is transitive and, in light of Theorem 3.7, \[ \operatorname{Stab}_{G}({gH}) = gHg^{-1}. \] Hence the kernel of this action is given by \[ \ker(G⟳G/H) = \bigcap_{g \in G} gHg^{-1}. \] This is called the normal core of $H$; it is the largest normal subgroup of $G$ contained in $H.$

Theorem 3.8 (Orbit-stabilizer) If $G ⟳X$ and $x \in G$, then \[ [G: \operatorname{Stab}_{G}({x})] = |\operatorname{Orb}_{G}({x})|. \]

Example 3.61 For $G ⟳G$ by left multiplication (Example 3.51), each stabilizer is trivial and so Theorem 3.8 gives the unimpressive tautology $|G| = |G|.$

Example 3.62 For $G ⟳G$ by conjugation (Example 3.52), applying Theorem 3.8 yields: \[ [G: \operatorname{C}_{G}({x})] = | \operatorname{cl}_{G}({x}) \, |. \]

Example 3.63 For $G ⟳G/H$ by left multiplication (Example 3.53), Theorem 3.8 to $eH \in G/H$ recovers Theorem 3.2.

Group Algebra

Definition 3.15 Fix a field $F$ and a finite group $G.$ Of great importance to this course is the group algebra, denoted $F[G]$, which is a $|G|$-dimensional $F$-algebra. The group algebra comes with a canonical basis, one element for every $g \in G$ which we denote $e_g \in F[G].$ We define multiplication in this ring by \[ e_g \cdot e_h \coloneqq e_{gh}. \] In other words, $F[G]$ is built from formal $F$-linear sums of the elements in $G = \{g_1,\dots,g_n\}$, \[ \alpha_{g_1} e_{g_1} + \cdots + \alpha_{g_n} e_{g_n} \in F[G], \] and inherits a notion of multiplication from the group law. In particular, note that $F[G]$ is a commutative $F$-algebra if and only if $G$ is abelian. The group algebra comes with a natural $G$ action, namely by extending $g \cdot e_h \coloneqq e_{gh}$ linearly.

Example 3.64 The group algebra $\mathbb{R}[\mathbb{Z}/{3}\mathbb{Z}]$ consists of vectors of the form $\alpha_0 e_0 + \alpha_1 e_1 + \alpha_2 e_2.$ We can write down the action $\mathbb{Z}/{3}\mathbb{Z} \to \operatorname{Aut}(\mathbb{R}[\mathbb{Z}/{3}\mathbb{Z}])$ explicitly: \[ \begin{gathered} 0 \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}, \\ 1 \mapsto \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{pmatrix}, \\ 2 \mapsto \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{pmatrix}. \end{gathered} \] These matrices all commute, as they are in the image of a homomorphism from an abelian group, so we hope that they might diagonalize simultaneously. Indeed, writing $\zeta = e^{2\pi i/3}$, we have: \[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix},\ \begin{pmatrix} 1 & 0 & 0 \\ 0 & \zeta & 0 \\ 0 & 0 & \zeta^2 \\ \end{pmatrix},\ \text{ and }\ \begin{pmatrix} 1 & 0 & 0 \\ 0 & \zeta^2 & 0 \\ 0 & 0 & \zeta \\ \end{pmatrix} \] after changing to the basis $\{ e_0+e_1+e_2, e_0 + \zeta^2 e_1 + \zeta e_2, e_0 + \zeta e_1 + \zeta^2 e_2 \}.$ Over $\mathbb{R}$, however, the best we can do is change to a basis like $\{ e_0+e_1+e_2, e_0-e_1, e_0+e_1-2e_2 \}$: \[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix},\ \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1/2 & -3/2 \\ 0 & 1/2 & -1/2 \\ \end{pmatrix},\ \text{ and }\ \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1/2 & 3/2 \\ 0 & -1/2 & -1/2 \\ \end{pmatrix} \] While not diagonal, these transformations have all been placed into a block diagonal form. Hence, we have obtained a vector space decomposition $\mathbb{R}[\mathbb{Z}/{3}\mathbb{Z}] = U \oplus V$, where \[ U = \operatorname{Span}\{e_0+e_1+e_2\} \text{ and } V = \operatorname{Span}\{e_0-e_1, e_0+e_1-2e_2 \}, \] that respects the action of $\mathbb{Z}/{3}\mathbb{Z}$; furthermore, we can see that the action of $\mathbb{Z}/{3}\mathbb{Z}$ on $U$ is trivial and the action on $V$ resembles rotations of the plane by increments of $120^\circ.$ Breaking actions down into more comprehensible pieces is the raison d’être of representation theory.

Some references, especially with geometric inclinations (e.g. Shurman (1997)), write $\mathcal{D}_{n}$ for this group—preferring to emphasize the number of vertices rather than the number of elements. We opt for the $2n$ notation for consistency with common sources.↩︎
As we will see, it is not generally correct to define an “isomorphism” as a “bijective homomorphism.” The correct definition should be: “bijective homomorphism whose inverse is also a homomorphism.” We are fortunate in many algebraic categories that these notions are equivalent.↩︎
Index is analogous to the notion of codimension (cf. Definition 2.12)↩︎
Some books, like Dummit and Foote (2003), write $G_x$ for $\operatorname{Stab}_{G}({x})$, but this notation is nearly indistinguishable from the orbit $\operatorname{Orb}_{G}({x}).$↩︎

\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)	\(12\)
\(\mathcal{C}_{1}\)	\(\mathcal{C}_{2}\)	\(\mathcal{C}_{3}\)	\(\mathcal{C}_{4}\)	\(\mathcal{C}_{5}\)	\(\mathcal{C}_{6}\)	\(\mathcal{C}_{7}\)	\(\mathcal{C}_{8}\)	\(\mathcal{C}_{9}\)	\(\mathcal{C}_{10}\)	\(\mathcal{C}_{11}\)	\(\mathcal{C}_{12}\)
			\(\mathcal{V}\)		\(\mathcal{S}_{3}\)		\(\mathcal{C}_{4} \times \mathcal{C}_{2}\)	\(\mathcal{C}_{3} \times \mathcal{C}_{3}\)	\(\mathcal{D}_{10}\)		\(\mathcal{C}_{6} \times \mathcal{C}_{2}\)
							\(\mathcal{V}\times \mathcal{C}_{2}\)				\(\mathcal{A}_{4}\)
							\(\mathcal{D}_{8}\)				\(\mathcal{D}_{12}\)
							\(\mathcal{Q}_{8}\)				\(\mathcal{Q}_{12}\)