Groups Lecture 3

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in which we meet group homomorphisms.

As a small add-on from last time, we started by proving that the subgroups of the integers \mathbb{Z} are exactly the sets of multiples n\mathbb{Z}. We then had a detour on functions f\colon {X\to Y} from domain (or source) X to codomain (or target) Y. We saw how to compose functions, when they are the same, and what injective, surjective and bijective functions are. Our examples included the identity function and the inclusion map, as well as a few others. We noted (but didn’t prove) that bijective functions have inverses f^{-1}\colon {Y\to X} with f\circ f^{-1}=1_Y and f^{-1}\circ f=1_X. Our main definition today was that of a group homomorphism: a function f\colon {G\to H} between two groups which preserves the group structure, meaning that f(a*b)=f(a)\star f(b) for all a,b\in G. This means it does not matter whether we first combine two elements in G and then send the result over to H, or whether we send the elements separately to H and combine them there. We call bijective group homomorphisms isomorphisms. Examples of homomorphisms include identity, inclusion of a subgroup, the exponential map \exp\colon {(\mathbb{R},+)\to (\mathbb{R}^+,\cdot)}, and the determinant of a 2\times 2 matrix.

Understanding today’s lecture

Practice Sheet C is the last one you’ll get from me, so you should get used to making easy exercises like that for yourself if you need them. Playing around with functions seeing if they are homomorphisms or not is very useful. And you can also keep asking yourself if you meet a function “is it injective, surjective, bijective?” Perhaps you might investigate what the difference is between \exp\colon {(\mathbb{R},+)\to (\mathbb{R}^+,\cdot)} and \exp\colon {(\mathbb{R},+)\to (\mathbb{R}^*,\cdot)}. Are they the same function/homomorphism or do they have different properties? Does it matter which one we use? Sheet C asks you something about a constant function: that is a function that sends all the elements in the domain to the same element in the codomain, i.e. everything gets mapped to the same single element.

Going a little deeper

The style of proof we saw for the fact that the subgroups of \mathbb{Z} are exactly n\mathbb{Z} is one we will meet again. It crucially uses the fact that any subset of natural numbers has a smallest element. This is a very special property of the natural numbers, and you will see more about it in Numbers and Sets, I expect. Look out for (or google) the Peano Axioms or why Proof by Induction works.

In modern mathematics, it is always important to know “what kind of functions belong to what kind of objects”. So when we study groups, one of the first questions we should ask ourselves is: what functions belong to groups, i.e. what functions preserve the group structure? We have asked a group homomorphism to preserve the group operation. But is that the only thing that makes a group a group? What about the identity and inverses? You might think about this before next time.

Preparing for Lecture 4:

Next time we will prove some  properties of group homomorphisms, and learn about images and kernels. These are special subgroups related to a group homomorphism. If you haven’t done the exercise “The composite of bijective functions is bijective.” yet, now would be a good time to try it. Can you think of some properties you would like group homomorphisms to have so that we can truly say “they preserve the group structure”?

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