Groups Lecture 2

in which we meet some properties of groups and define subgroups.

In our second lecture we proved some simple properties of groups, such as the uniqueness of the identity and inverses, the fact that any element is the inverse of its inverse, and the “socks and shoes” lemma about inverses of a product. We defined the concept of a subgroup, being a subset which is also a group with the same operation, and saw two subgroup criteria (the “usual” subgroup criterion and the “lazy” subgroup criterion) that are useful in checking whether something is a subgroup or not. We saw a few examples of subgroups, mainly referring back to examples we saw in Lecture 1.

Understanding today’s lecture

Practice Sheet B is a good starting point to get used to proving general results using just the group axioms. I especially recommend that you spend a bit of time on question 3, about associativity when we try to multiply more than three elements. See whether you like the “usual” or the “lazy” subgroup criterion better when showing something is a subgroup. Remember, for the “lazy” one you do need to show that the subset is non-empty, otherwise it all falls apart!

Going a little deeper

I find the fact that the groups axioms are strong enough to force the identity in any subgroup to be the same element as in the bigger group quite amazing. Of course, when we say we want a subgroup, it makes sense to expect that everything we have in the big group restricts to the smaller group: same group operation, same identity, same inverses for those elements which are in the smaller group. However, we don’t have to ask this, but it follows automatically from the fact that we just ask the subset to be  a group with the same operation! The inverses are doing the magic for us here: if you have something like a group but without asking every element to have an inverse, this is not necessarily true. A set with an operation satisfying closure, associativity and having an identity is called a monoid. For example taking $\mathbb{Z}_6$, the integers mod $6$, with multiplication gives us a monoid. The identity is clearly $1$. Then if we take the subset $2\mathbb{Z}_6=\{0,2,4\}$, we still get a monoid with multiplication, but with a different identity: it is now $4$ because $4*4\equiv 4 \mod 6$ and $4*2 \equiv 2 \mod 6$ (and trivially $4*0\equiv 0 \mod 6$).

Preparing for Lecture 3:

Next time we will think about how we can “get from one group to another” without destroying all the structure. We will define “structure-preserving functions”, called group homomorphisms. Perhaps you can already work out what a possible definition might be? And when would you say two groups are “the same enough” for us not to care too much about which one we look at?