Thread: subgroups question

Could you help me with this question... I've been looking at it for hours and don't know what to do...

i) A subgroup 'H' of the symmetric group S4 consists of the permutations id, (1 2)(3 4), (1 3)(2 4) and 'a'. Determine the permutation 'a'.

ii) Is 'H' cyclic?

Originally Posted by daaavo

Could you help me with this question... I've been looking at it for hours and don't know what to do...

i) A subgroup 'H' of the symmetric group S4 consists of the permutations id, (1 2)(3 4), (1 3)(2 4) and 'a'. Determine the permutation 'a'.

Well its just ((1 2)(3 4))((1 3)(2 4)) = (1 4)(2 3).

ii) Is 'H' cyclic?

No. Observe that [(1 2)(3 4)]^2 = [(1 3)(2 4)]^2 = [(1 4)(2 3)]^2 = 1. There is no element of order greater than 2 and thus H cant be cyclic.

In fact $\displaystyle H \cong C_2 \times C_2$

thanks for that... I'm just wondering, why is it that H cannot be cyclic because there is no element of order greater than 2?

Originally Posted by daaavo

thanks for that... I'm just wondering, why is it that H cannot be cyclic because there is no element of order greater than 2?

A cyclic group is generated by one element, you have 4 elements in your group you there must be a element of order 4 in the group to generate all of them.