2.4 Homomorphisms and isomorphisms

Homomorphisms

Let $G$ and $G^{\prime }$ be groups. A homomorphism $\phi :G\to G^{\prime }$ is a map from $G$ to $G^{\prime }$ tha preserves products,

$$\phi (ab)=\phi (a)\phi (b)\text{ for all }a,b\in G.$$

Theorem 2.1

Under a homomorphism $\phi :G\to G^{\prime }$ the identity e ofG is mapped to the identity $e^{\prime }of{G}^{\prime }$ , and the inverse ofany element g ofG to the inverse ofits image $\phi (g)$

Proof

For any $g\in G$

$$\phi (g)=\phi (ge)=\phi (g)\phi (e).$$

Multiplying both sides of this equation on the left by $(\phi (g){)}^{-1}$ gives the desired result,

$$e^{\prime }=e^{\prime }\phi (e)=\phi (e).$$

If $g\in G$ then $\phi (g^{-1})\phi (g)=\phi (g^{-1}g)=\phi (e)=e^{\prime }$ . Hence $\phi (g^{-1})=(\phi (g){)}^{-1}$ as required. -

Exercise

If $\phi :G\to G^{\prime }$ is a homomorphism, show that the image set

$$\mathrm{im}(\phi )=\phi (G)=\{g^{\prime }\in G^{\prime }|g^{\prime }=\phi (g),g\in G\}\text{\hspace{1em}(2.13)}$$

is a subgroup of $G^{\prime }.$

Example 2.16

For any real number $x\in \mathbb{R}$ , define its integral part [x] to be the largest integer that is less than or equal to x, and itsfractional part to be $(x)=x-[x]$ . Evidently $0\le (x)<1$ . On the half-open interval [0, 1) of the real line define addition modulo 1 by

$$a+b\mathrm{mod}1=(a+b)=\text{ fractional part of }a+b.$$

This defines an abelian group, the group of real numbers modulo 1. To verify the group axioms we note that 0 is the identity element and the inverse of any $a>0$ is $1-a$ . The inverse of $a=0$ is 0.

The map ${\phi }_1:\mathbb{R}\to [0,1)$ from the additive group of real numbers to the group of rea numbers modulo 1 defined by ${\phi }_1(x)=(x)$ is a homomorphism since $(x)+(y)=(x+y)$

Exercise

Show that the circle map or phase map $C:\dot{\mathbb{C}}\to [0,2\pi )$ defined by

$$C(z)=\theta \text{ where }z=|z|{\mathrm{e}}^{i\theta },0\le \theta <2\pi$$

is a homomorphism from the multiplicative group of complex numbers to the additive group of real modulo 2π, defined in a similar way to the reals modulo 1 in the previous example

Example 2.17

Let sign : $S_n\to \{+1,-1\}$ be the map that assigns to every permutation π its parity,

$$\mathrm{sign}(\pi )=(-1{)}^{\pi }=\{\begin{array}{ll}+1& \text{ if }\pi \text{ is even },\\ -1& \text{ if }\pi \text{ is odd }.\end{array}$$

From Eq. (2.1), sign is a homomorphism from $S_n$ to the multiplicative group of reals

$$\mathrm{sign}(\pi \sigma )=\mathrm{sign}(\pi )\mathrm{sign}(\sigma ).$$

Exercise

From Eq. (2.5) show that the determinant map det : $GL(n,\mathbb{R})\to$ R˙ from the general linear group of order n to the multiplicative group of reals is a homomorphism.

Isomorphisms

An isomorphism is a homomorphism that is one-to-one and onto. Ifan isomorphism exists between two groups $G$ and $G^{\prime }$ they are said to be isomorphic, written $G\cong G^{\prime }$ . The two groups are then essentially identical in all their group properties.

Exercise

Show that if $\phi :G\to G^{\prime }$ is an isomorphism, then so is the inverse map ${\phi }^{-1}:G^{\prime }\to G$

Exercise

I $\cdot \phi :G\to G^{\prime }$ and $\psi :G^{\prime }\to G^{\prime \prime }$ are isomorphisms then so is $\psi \circ \phi :G\to G^{\prime \prime }$

These two statements show that isomorphism is a symmetric and transitive relation on the class of all groups. Hence it is an equivalence relation on the class of groups, since the reflexive property follows from the fact that the identity map ${\mathrm{id}}_G:G\to G$ is trivially an isomorphism. Note that the word ‘class’ must be used in this context because the ‘set of all groups’ is too large to be acceptable. Group theory is the study of equivalence classes of isomorphic groups. Frequently it is good to single out a special representative of an equivalence class. Consider, for example, the following useful theorem for finite groups:

Theorem 2.2

(Cayley) Every finite group $G$ of order n is isomorphic to a permutation group.

Proof

For every $g\in G$ define the map $L_g:G\to G$ to be left multiplication by $g,$

$$L_g(x)=gx\text{ where }x\in G.$$

This map is one-to-one and onto since

$$gx=g{x}^{\prime }⟹x=x^{\prime }\text{ and }x=L_g(g^{-1}x)\text{ for all }x\in G.$$

The map $L_g$ therefore permutes the elements of $G=\{g_1=e,g_2,...,g_n\}$ and may be identified with a member of $S_n$ . It has the property $L_g\circ L_h=L_{gh}$ , since

$$L_g\circ L_h(x)=g(hx)=(gh)x=L_{gh}(x),\forall x\in G.$$

Hence the map $\phi :G\to S_n$ defined by $\phi (g)=L_g$ is a homomorphism,

$$\phi (g)\phi (h)=L_g\circ L_h=L_{gh}=\phi (gh).$$

Furthermore, $\phi$ is one-to-one, for if $\phi (g)=\phi (h)$ then $g=L_g(e)=L_h(e)=h$ . Thus $G$ is isomorphic to the subgroup of $\phi (G)\subseteq S_n$ -

From the abstract point of view there is nothing to distinguish two isomorphic groups, but different ‘concrete’ versions of the same group may have different applications. The particular concretization as linear groups of transformations or matrix groups is known as group representation theory and plays a major part in mathematical physics.

Automorphisms and conjugacy classes

An automorphism is an isomorphism $\phi :G\to G$ of a group onto itself. A trivial example is the identity map ${\mathrm{id}}_G:G\to G$ . Since the composition of any pair of automorphisms is an automorphism and the inverse of any automorphism ${\phi }^{-1}$ is an automorphism, it follows that the set of all automorphisms of a group G is itself a group, denoted $\mathrm{Aut}(G)$

If g is an arbitrary element of $G$ , the map $C_g:G\to G$ defined by

$$C_g(a)=ga{g}^{-1}\text{\hspace{1em}(2.14)}$$

is called conjugation by the element $g.$ . This map is a homomorphism, for

$$C_g(ab)=gab{g}^{-1}=ga{g}^{-1}gb{g}^{-1}=C_g(a)C_g(b),$$

and $C_{g^{-1}}$ is its inverse since

$$C_{g^{-1}}\circ C_g(a)=g^{-1}(ga{g}^{-1})g=a,\forall a\in G.$$

Hence every conjugation $C_g$ is an automorphism of $G$ . Automorphisms that are a conjuga tion by some element $g$ of $G$ are called inner automorphisms. The identity $C_{gh}=C_g\circ C_h$ holds, since for any $a\in G$

$$C_{g}h(a)=gha(gh{)}^{-1}=gha{h}^{-1}{g}^{-1}=C_g\left(C_h(a)\right).$$

Hence the map $\psi :G\to \mathrm{Aut}(G)$ , defined by $\psi (g)=C_g,$ , is a homomorphism. The inner automorphisms, being the image of $G$ under $\psi$ , form a subgroup of $\mathrm{Aut}(G)$ ). Two subgroups H and $H^{\prime }$ of $G$ that can be transformed to each other by an inner automorphism of $G$ are called conjugate subgroups. In this case there exists an element $g\in G$ such that

$$H^{\prime }=gH{g}^{-1}=\{gh{g}^{-1}\mid h\in H\}.$$

Exercise

Show that conjugacy is an equivalence relation on the set of all subgroups of a group $G.$ What is the equivalence class containing the trivial subgroup $\{e\}2$

Conjugation also induces an equivalence relation on the original group $G$ by $a\equiv b$ ifand only if there exists $g\in G$ such that $b=C_g(a)$ . The three requirements for an equivalence relation are easily verified: (i) reflexivity, $a=C_e(a)$ for all $a\in G;(\mathrm{ii})$ ) symmetry, if $b=$ $C_g(a)$ then $a=C_{g^{-1}}(b)$ ; (iii) transitivity, i $\dot{b}=C_g(a)$ ) and $c=C_h(b)$ then $c=C_{hg}(a)$ . The equivalence classes with respect to this relation are called conjugacy classes. The conjugacy class of an element $a\in G$ is denoted $G_a$ . For example, the conjugacy class of the identity is always the singleton $G_e=\{e\}$ , since $C_{g}e=ge{g}^{-1}=e$ for all $g\in G$

Exercise

What are the conjugacy classes of an abelian group?

Example 2.18

For a matrix group, matrices A and B in the same conjugacy class are related by a similarity transformation

$$\mathsf{B}={\mathsf{SAS}}^{-1}.$$

Matrices related by a similarity transformation have identical invariants such as determinant, trace (sum of the diagonal elements) and eigenvalues. To show determinant is an invariant use Eq. (2.5),

$$\det \mathsf{B}=\det \mathsf{S}\det \mathsf{A}(\det \mathsf{S}{)}^{-1}=\det \mathsf{A}.$$

For the invariance of trace we need the identity

$$\mathrm{tr}(\mathsf{AB})=\mathrm{tr}(\mathsf{BA}),\text{\hspace{1em}(2.15)}$$

which is proved by setting $\mathsf{A}=[a_{ij}]$ and $\mathsf{B}=[b_{ij}]$ and using the multiplication law o matrices,

$$\mathrm{tr}(\mathsf{AB})=\sum _{i=1}^n\left(\sum _{j=1}^n{a}_{ij}{b}_{ji}\right)=\sum _{j=1}^n\left(\sum _{i=1}^n{b}_{ji}{a}_{ij}\right)=\mathrm{tr}(\mathsf{BA}).$$

Hence

$$\mathrm{tr}\mathsf{B}=\mathrm{tr}({\mathsf{SAS}}^{-1})=\mathrm{tr}({\mathsf{S}}^{-1}\mathsf{SA})=\mathrm{tr}(\mathsf{IA})=\mathrm{tr}\mathsf{A},$$

as required. Finally, if λ is an eigenvalue of A corresponding to eigenvector v, then $\mathsf{S}\mathbf{v}$ is an eigenvector of B with the same eigenvalue,

$$\mathsf{Av}=\lambda \mathsf{v}⟹\mathsf{B}(\mathsf{Sv})={\mathsf{SAS}}^{-1}\mathsf{Sv}=\mathsf{SAv}=\lambda \mathsf{Sv}.$$

Example 2.19

The conjugacy classes of the permutation group $S_3$ are, in cyclic notation,

$$\{e\};\{(12),(13),(23)\};\text{ and }\{(123),(132)\}.$$

These are easily checked by noting that $(12{)}^{-1}(123)(12)=(132)$ and

$$(123{)}^{-1}(12)(123)=(13),\text{ etc. }$$

It is a general feature of permutation groups that conjugacy classes consist of permutations having identical cycle structure (see Problem 2.11).

Problems

Problem 2.9

Show that Theorem 2.2 may be extended to infinite groups as well. That is, any group G is isomorphic to a subgroup of Transf(G), the transformation group of the set G.

Problem 2.10

Find the group multiplication tables for all possible groups on four symbols $e,a,b$ and $c,$ and show that any group of order 4 is either isomorphic to the cyclic group ${\mathbb{Z}}_4$ or the product group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$

Problem 2.11

Show that every cyclic permutation $(a_1{a}_2\dots a_n)$ has the property that for any per mutation π,

$$\pi (a_1{a}_2\dots a_n){\pi }^{-1}$$

is also a cycle of length n. [Hint: It is only necessary to show this for interchanges $\pi =(b_1{b}_2)$ ) as every permutation is a product of such interchanges.]

(a) Show that the conjugacy classes of $S_n$ consist of those permutations having the same cycle structure, e.g. (1 2 3)(4 5) and $(146)(23)$ belong to the same conjugacy class.

(b) Write out all conjugacy classes of $S_4$ and calculate the number of elements in each class.

Problem 2.12

Show that the class of groups as objects with homomorphisms between groups as morphisms forms a category – the category ofgroups (see Section 1.7). What are the monomorphisms, epimorphisms and isomorphisms of this category?