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.