find a compostion series?

Question

find a composition series for a Sylow 2-subgroup of S8.

note: S8 is S subscript 8.

Answer 1

I am not certain I understand the question, but for S8 any disjoint set of 2-cycles is a permutation that generates a subgroup of order 2, such as for example (12), or (35)(48)(17).

Answer 2

I just looked at Herstein regarding this problem. Theorem 2.12.1 says that there are subgroups of all orders p^k where p^k divides the order of the group.

Our group is of order 2^7. It has a subgroup of order 2^6. The nbr of cosets is 2, so it is normal and the quotient group is simple (Z2). That subgroup has a subgroup of order 2^5, etc.This makes a composition series for the group.