# Group under composition

Which proper subsets of $S_3$ for a group under composition?

I'm not really sure how to approach this. I know the four requirements of groups - identity, closure, inverse and associativity.

And I know that $S_3$ comprises:

(abc) (acb) (bac) (bca) (cba) (cab)

How would I begin to identify the subsets here that meet those four requirements? I would appreciate some hints and tips. I don't just want an answer...I need to actually understand how to answer these types of questions.

• There are 64 subsets of $S_3$, but only 32 of them contain the identity element. – Jasha Feb 2 '15 at 3:21

A generic strategy is to try to put an element in the set, and then take products to "close" the set. So for instance, you will find that any time $(bca)$ is in the set, $(bca)(bca)=(cab)$ must be in as well.
• Given a group $G$, each subset $T$ of $G$ determines a unique subgroup of $G$ called the subgroup generated by $T$. This subgroup, often denoted $\langle T\rangle$, is defined as the smallest subgroup of $G$ which contains $T$ as a subset. – Jasha Feb 2 '15 at 3:44
• @Eric Stucky - I may have failed to mention this but I am asked to find proper subgroups. This is why I have omitted $S_3$ itself from the list given in my previous comment. With this knowledge, do you consider the previous list complete? Or am I still missing something obvious? – 123 Feb 2 '15 at 17:37