Groups and Group Actions: Lecture 9

In which we think about homomorphisms, and wonder how many genuinely different groups there are with small orders.

Definitions of a homomorphism, isomorphism and automorphism.
Proposition 41: Let , be groups, let $\theta : G \to H$ be a homomorphism. Take $g \in G$ , $n \in \mathbb{Z}$ . Then
1. $\theta(e_G) = e_H$ ;
2. $\theta(g^{-1}) = [\theta(g)]^{-1}$ ; and
3. $\theta(g^n) = [\theta(g)]^n$ .
The first followed by considering $\theta(e_G \ast_G e_G)$ in two ways. The second followed from $e_H = \theta(e_G) = \theta(g g^{-1})$ . The third falls out using induction and previous parts.
Corollary 42: Let , be groups, let $\theta : G \to H$ be a homomorphism. Take $g \in G$ of finite order. Then $o(\theta(g))$ divides . Moreover, if $\theta$ is an isomorphism then $o(\theta(g)) = o(g)$ . We showed that $[\theta(g)]^{o(g)} = e_H$ , and then the first part follows using Lemma 23. For the second part, we noted that for injective $\theta$ we have $[\theta(g)]^k = e_H$ if and only if .
Lemma 43: Let be a finite group with even order. Then contains an element of order . We defined a relation $\sim$ on via $x \sim y$ if and only if or $x = y^{-1}$ , and noted that this is an equivalence relation. The equivalence classes have size 1 ( $\{g\}$ for elements that are self-inverse — ) or 2 ( $\{g, g^{-1}\}$ ). By counting the number of each, and remembering that equivalence classes partition the set, we saw that the number of classes of size 1 is even. Since it’s also at least 1, there must be a non-identity element that’s self-inverse.
Theorem 44: Let be an odd prime. Let be a group with order . Then is isomorphic to $C_{2p}$ or $D_{2p}$ . If is cyclic, then is isomorphic to $C_{2p}$ , so we supposed that contains no element of order . We showed that contains an element of order 2, and another of order , and then $G = \{e, y, y^2, \dotsc, y^{p-1}, x, xy, xy^2, \dotsc, xy^{p-1}\}$ . By considering the order of , we found that $yx = xy^{-1}$ , so is isomorphic to $D_{2p}$ .
Definition of a quaternion.
Definition of the kernel and image of a homomorphism.

Understanding today’s lecture

It would be good to practise writing out some checks that suitable functions are homomorphisms. Pick two groups, can you find a homomorphism (an interesting homomorphism!) between them? How does Corollary 42 help you to narrow down the possibilities?

If you found the proof of Theorem 44 a bit daunting, then a really good plan would be to work through it in a specific case (eg groups of order ) to see what it says — this is a great way to get insight into a proof.

Can you check that Q_8 really is a group?

Preparation for Lecture 10

What are the homomorphisms from $\mathbb{Z}$ to itself?

Can you say anything interesting about the kernel and image of a homomorphism as subsets of their respective groups? (Hint: what can you say about the kernel and image of a linear map?)

We can define a binary operation on (left) cosets of a subgroup in a group , by defining $(g_1 H) \ast (g_2 H) = (g_1 g_2)H$ for all g_1 , $g_2 \in G$ . Is this well defined? That is, if we pick g_1' and g_2' with g_1 H = g_1'H and g_2 H = g_2'H , do we necessarily have (g_1 g_2) H = (g_1' g_2') H ? What conditions do we need to impose on to make this work?