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.

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

Not sure if this is a full answer to the question, but the requirement you're going to run up against will always be closure (and inverses, but for finite groups this is a special case).

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.

If you know Lagrange's theorem you can limit your search space considerably.

  • $\begingroup$ 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. $\endgroup$ – Jasha Feb 2 '15 at 3:44
  • $\begingroup$ @Eric Stucky - I've read over Lagrange's theorem and it did help me considerably. I searched through the sets with 1,2 and 3 elements. It seems like e, e and any two cycle and e and both 3 cycles form the subgroups. Is this correct? $\endgroup$ – 123 Feb 2 '15 at 4:22
  • $\begingroup$ @mathtastic: That is true. By Lagrange's Theorem you are almost done: there is one more (obvious) one that you're missing. $\endgroup$ – Eric Stucky Feb 2 '15 at 7:08
  • $\begingroup$ @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? $\endgroup$ – 123 Feb 2 '15 at 17:37

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