Find all the subgroups of a subgroup of $S_4$

Question

I have $H=\langle \ (1\ 2\ 3 4), \ (1 \ 2)(3 \ 4) \ \rangle $, and I need to find all the subgroups of $H$.

I have found the elements of $H$ and their order.

Also I have found (by using Lagrange's Theorem) that these subgroups must have order 1,2,4 or 8.

And lastly I have found the possible order 1,2,8 subgroups.

My question is are there any other theorems (like Lagrange's) or clever tricks that I can use to narrow down my work in finding the order 4 subgroups?

Because the best idea I've got now is to try $\{I,x,y,z\}$ where $I$ is the identity element and $x,y,z \in H$, and this looks messy and unordered.

Thanks.

Any subgroup of order $2$ will either be cyclic (so you are looking for elements of order $4$), or generated by two commuting elements of order $2$. — Tobias Kildetoft, Feb 26 '16 at 08:34
Ohh, just noticed the typo in the above comment, I of course meant subgroup of order $4$. — Tobias Kildetoft, Feb 26 '16 at 10:00

score 2 · Answer 1 · answered Feb 26 '16 at 09:49

Observe that

$$(12)(34)^2=(1234)^4=1\;,\;\;(12)(34)(1234)(12)(34)=(1432)=(1234)^{-1}$$

so that in fact $\;H\cong D_4=\,$ the dihedral group of order $\;8\;$ , and from here you can deduce all its subgroups (five subgroups of order two, three of order four and both trivial subgroups)

score 0 · Answer 2 · edited Apr 13 '17 at 12:20

0

If you are interested in solve it using any software you could use GAP, you can check How to find all subgroups of a group in GAP in order to solve your question.

edited Apr 13 '17 at 12:20

Community

1

answered Feb 26 '16 at 08:51

Angel Blasco

566

3

I think this is terrible advice for the case at hand. The exercise is meant to make you familiar with the groups in question and their subgroups and how to work with them. By hand. – Tobias Kildetoft Feb 26 '16 at 09:45

Find all the subgroups of a subgroup of $S_4$

2 Answers2