2.4: Group Homomorphisms

Definition 2.4.1. Group homomorphism.

Let $G,H$ be groups.

A map $\phi\colon G\to H$ is called a homomorphism if

$$\phi(xy) = \phi(x)\phi(y) \nonumber$$

for all $x,y$ in $G\text{.}$ A homomorphism that is both injective (one-to-one) and surjective (onto) is called an isomorphism of groups. If $\phi\colon G\to H$ is an isomorphism, we say that $G$ is isomorphic to $H\text{,}$ and we write $G\approx H\text{.}$

Checkpoint 2.4.2.

Show that each of the following are homomorphisms.

• $GL(n,\mathbb{R})\to \mathbb{R}^\ast$ given by $M\to \det M$
• $\mathbb{Z}\to \mathbb{Z}$ given by $x\to mx\text{,}$ some fixed $m\in \mathbb{Z}$
• $G\to G\text{,}$ $G$ any group, given by $x\to axa^{-1}\text{,}$ some fixed $a\in G$
• $\mathbb{C}^\ast\to\mathbb{C}^\ast$ given by $z\to z^2$

Show that each of the following are not homomorphisms. In each case, demonstrate what fails.

• $\mathbb{Z}\to \mathbb{Z}$ given by $x\to x+3$
• $\mathbb{Z}\to \mathbb{Z}$ given by $x\to x^2$
• $D_4\to D_4$ given by $g\to g^2$

Definition 2.4.3. Kernel of a group homomorphism.

Let $\phi\colon G\to H$ be a group homomorphism, and let $e_H$ be the identity element for $H\text{.}$ We write $\ker(\phi)$ to denote the set

$$\ker(\phi) :=\phi^{-1}(e_H) = \{g\in G\colon \phi(g)=e_H\}, \nonumber$$

called the kernel of $\phi\text{.}$

Checkpoint 2.4.4.

Find the kernel of each of the following homomorphisms.

• $\mathbb{C}^\ast\to \mathbb{C}^\ast$ given by $z\to z^n$
• $\mathbb{Z}_8\to \mathbb{Z}_8$ given by $x\to 6x \pmod{8}$
• $G\to G\text{,}$ $G$ any group, given by $x\to axa^{-1}\text{,}$ some fixed $a\in G$

Answer

1. $\displaystyle C_n$
2. $\displaystyle \langle 4\rangle = \{0,4\}$
3. $\displaystyle \{x\in G\colon axa^{-1}=e\}=C(a)$

Proposition 2.4.5. Basic properties of homomorphisms.

Let $\phi\colon G\to H$ be a homomorphism of groups. Let $e_G,e_H$ denote the identity elements of $G,H\text{,}$ respectively. We have the following.

1. (identity goes to identity) $\phi(e_G) = e_H$
2. (inverses go to inverses) $\phi\left(g^{-1}\right) = \left(\phi(g)\right)^{-1}$ for all $g\in G$
3. $\ker(\phi)$ is a subgroup of $G$
4. $\phi(G)$ is a subgroup of $H$
5. (preimage sets are cosets of the kernel) $\phi(x)=y$ if and only if $\phi^{-1}(y) = x\ker(\phi)$
6. $\phi(a)=\phi(b)$ if and only if $a\ker(\phi)=b\ker(\phi)$
7. $\phi$ is one-to-one if and only if $\ker(\phi)=\{e_G\}$

Proof.

See Exercise Group 2.4.2.1–3.

Proposition 2.4.6. $G/K$is a group if and only if $K$ is a kernel.

Let $K$ be a subgroup of a group $G\text{.}$ The set $G/K$ of cosets of $K$ forms a group, called a quotient group (or factor group), under the operation

$$(xK)(yK) = xyK\label{multkernelcosets}\tag{2.4.1}$$

if and only if $K$ is the kernel of a homomorphism $G\to G'$ for some group $G'\text{.}$

Proof.

See Exercise 2.4.2.5.

Here is a corollary of Proposition 2.4.6 and its proof.

Corollary 2.4.7. (First Isomorphism Theorem).

Let $\phi\colon G\to H$ be a homomorphism of groups. Then $G/\ker(\phi)$ is isomorphic to $\phi(G)$ via the map $g\ker(\phi) \to \phi(g)\text{.}$

Definition 2.4.8. Normal subgroup.

A subgroup $H$ of a group $G$ is called normal if $ghg^{-1}\in H$ for every $g\in G\text{,}$ $h\in H\text{.}$ We write $H\trianglelefteq G$ to indicate that $H$ is a normal subgroup of $G\text{.}$

Proposition 2.4.9. Characterization of normal subgroups.

Let $K$ be a subgroup of a group $G\text{.}$ The following are equivalent.

1. $K$ is the kernel of some group homomorphism $\phi\colon G\to H$
2. $G/K$ is a group with multiplication given by Equation (2.4.1)
3. $K$ is a normal subgroup of $G$

Exercises

Exercise 1

Basic properties of homomorphism. Prove Proposition 2.4.5.

Prove Properties $1$ and $2$.

Prove Properties $3$ and $4$.

Prove Properties $5, 6,$ and $7$.

Hint

Use Fact 1.4.3.

Exercise2

Show that the inverse of an isomorphism is an isomorphism.

Exercise 3

Prove Proposition 2.4.6.

Exercise 4

Let $n,a$ be relatively prime positive integers. Show that the map $\mathbb{Z}_n\to \mathbb{Z}_n$ given by $x\to ax$ is an isomorphism.

Hint

Use the fact that $\gcd(m,n)$ is the least positive integer of the form $sm+tn$ over all integers $s,t$ (see Exercise 2.3.2.4). Use this to solve $ax=1 \pmod{n}$ when $a,n$ are relatively prime.

Exercise 5

Another construction of $\mathbb{Z}_n$.

Let $n\geq 1$ be an integer and let $\omega=e^{i2\pi/n}\text{.}$.ω=ei2π/n. Let $\phi\colon \mathbb{Z}\to S^1$ be given by $k\to \omega^k\text{.}$

1. Show that the the image of $\phi$ is the group $C_n$ of nth roots of unity.
2. Show that $\phi$ is a homomorphism, and that the kernel of $\phi$ is the set $n\mathbb{Z}=\{nk\colon k\in \mathbb{Z}\}\text{.}$
3. Conclude that $\mathbb{Z}/\!(n\mathbb{Z})$ is isomorphic to the group of $n$-th roots of unity.

Exercise 6: Isomorphic images of generators are generators

Let $S$ be a subset of a group $G\text{.}$ Let $\phi\colon G\to H$ be an isomorphism of groups, and let $\phi(S)=\{\phi(s)\colon s\in S\}\text{.}$ Show that $\phi(\langle S\rangle)=\langle \phi(S)\rangle\text{.}$

Exercise 7: Conjugation

Let $G$ be a group, let a be an element of $G\text{,}$ and let $C_a\colon G\to G$ be given by $C_a(g)=aga^{-1}\text{.}$ The map $C_a$ is called conjugation by the element $a$ and the elements $g,aga^{-1}$ are said to be conjugate to one another.

1. Show that $C_a$ is an isomorphism of $G$ with itself.
2. Show that "is conjugate to" is an equivalence relation. That is, consider the relation on $G$ given by $x\sim y$ if $y=C_a(x)$ for some $a\text{.}$ Show that this is an equivalence relation.

Exercise 8: Isomorphism induces an equivalence relation

Prove that "is isomorphic to" is an equivalence relation on groups. That is, consider the relation $\approx$ on the set of all groups, given by $G\approx H$ if there exists a group isomorphism $\phi\colon G\to H\text{.}$ Show that this is an equivalence relation.

Characterization of normal subgroups.

Prove Proposition 2.4.9. That Item 1 is equivalent to Item 2 is established by Proposition 2.4.6.

Exercise 9

Show that Item 1 implies Item 3.

Exercise 10

Show that Item 3 implies Item 2. The messy part of this proof is to show that multiplication of cosets is well-defined. This means you start by supposing that $xK=x'K$ and $yK=y'K\text{,}$ then show that $xyK=x'y'K\text{.}$

Exercise 11: Further characterizations of normal subgroups

Show that Item 3 is equivalent to the following conditions.

1. $gKg^{-1}= K$ for all $g\in G$
2. $gK = Kg$ for all $g\in G$

Exercise 12: Automorphisms

Let $G$ be a group. An automorphism of $G$ is an isomorphism from $G$ to itself. The set of all automorphisms of $G$ is denoted $Aut(G)$.

1. Show that $Aut(G)$ is a group under the operation of function composition.
2. Show that

   $$Inn(G) := \{C_g\colon g\in G\} \nonumber$$

   is a subgroup of $Aut(G)\text{.}$ (The group $Inn(G)$ is called the group of inner automorphisms of $G\text{.}$)

3. Find an example of an automorphism of a group that is not an inner automorphism.