Home Online Mathematical Communication Theorem of the week Groups and Group Actions: Lecture 10

Groups and Group Actions: Lecture 10

Theorem of the week 2018-03-12

In which we think some more about homomorphisms, and meet normal subgroups.

Proposition 45: Let $\theta : \mathbb{Z} \to \mathbb{Z}$ be a homomorphism. Then there is $n \in \mathbb{Z}$ with $\theta(m) = nm$ for all $m \in \mathbb{Z}$ . We proved this by defining $n = \theta(1)$ and then using the fact that $1$ generates $\mathbb{Z}$ .
Proposition 46: Let $G$ , $H$ be groups, let $\theta : G \to H$ be a homomorphism. Then
1. $\ker\theta$ is a subgroup of $G$ ; and
2. $\mathrm{Im} \theta$ is a subgroup of $H$ .
We proved this using the subgroup test.
Proposition 47: Let $G$ , $H$ be groups, let $\theta : G \to H$ be a homomorphism. Then $\theta$ is constant on each coset of $\ker\theta$ , and takes different values on different cosets. We saw that $\theta(g_1) = \theta(g_2)$ if and only if $g_1 g_2^{-1} = \ker\theta$ , and then used the coset equality test to see that this is equivalent to $g_1 \ker\theta = g_2 \ker \theta$ .
Corollary 48: Let $G$ , $H$ be groups, let $\theta : G \to H$ be a homomorphism. Then $\theta$ is injective if and only if $\ker\theta = \{e_G\}$ . This was immediate from Proposition 47, since $e_G \in \ker\theta$ .
Definition of a normal subgroup of a group.
Definition of a simple group.
Proposition 49: Let $G$ , $H$ be groups, let $\theta : G \to H$ be a homomorphism. Then $\ker\theta$ is a normal subgroup of $G$ . We already know from Proposition 46 that $\ker\theta$ is a subgroup of $G$ , so we just checked that if $k \in \ker\theta$ and $g \in G$ then $g^{-1} kg \in \ker\theta$ .
Proposition 50: Let $H$ be a subgroup of a group $G$ with index $|G/H|=2$ . Then $H \trianglelefteq G$ . We argued that the only (left and right) cosets of $H$ in $G$ are $H$ and $G\setminus H$ , and used this to see that $gH = Hg$ for all $g \in G$ .
Definition of conjugacy classes.

Understanding today’s lecture

Pick some homomorphisms. Can you identify their kernels and images? Which homomorphisms are injective? Which are surjective?

Pick a homomorphism $\theta$ between two groups (pick explicit groups and an explicit homomorphism). What are the cosets of $\ker \theta$ ? This might help you to get a feel for Proposition 47.

Can you prove Corollary 48 directly from the definitions, without using Proposition 47?

Pick some subgroups of groups. Which are normal in their respective groups?

Further reading

Of course Wikipedia has a page about normal subgroups, and another about the notion of a simple group. You will learn more about simple groups if you choose to study Group Theory next year.

MacTutor suggests that Galois was the first to recognise the significance of normal subgroups.

Preparation for Lecture 11

Can you show that if $H$ is a normal subgroup of $G$ then $G/H$ (the set of left cosets of $H$ in $G$ ) forms a group under the natural operation?