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 be a homomorphism. Take , . Then
- ; and
The first followed by considering in two ways. The second followed from . The third falls out using induction and previous parts.
- Corollary 42: Let , be groups, let be a homomorphism. Take of finite order. Then divides . Moreover, if is an isomorphism then . We showed that , and then the first part follows using Lemma 23. For the second part, we noted that if is injective then if and only if .
- Lemma 43: Let be a finite group with even order. Then contains an element of order . We defined a relation on via if and only if or , and noted that this is an equivalence relation. The equivalence classes have size 1 ( for elements that are self-inverse — ) or 2 (). 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 or . If is cyclic, then is isomorphic to , so we supposed that contains no element of order . We showed that contains an element of order 2, and another of order , and then . By considering the order of , we found that , so is isomorphic to .
- Definition of a quaternion.
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?
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 (e.g. groups of order ) to see what it says — this is a great way to get insight into a proof.
Can you check that really is a group?
Wikipedia has a page that lists lots of small groups.
The quaternion group can be described in many interesting ways. Quaternions were first described by Hamilton, who supposedly was so excited by his discovery that he carved the definition on a bridge in Dublin that he happened to be standing on at the time.
Preparation for Lecture 10
What are the homomorphisms from to itself?
Based on your experience of linear maps between vector spaces, how would you define the kernel and image of a homomorphism? Can you say anything interesting about them 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 for all , . Is this well defined? That is, if we pick and with and , do we necessarily have ? What conditions do we need to impose on to make this work?