Find the no. of distinct left cosets of H = { e, (16), (23), (16)(23)} in H6.

Hi anyone knows how to solve this?

find the number of distinct left cosets of H ={e, (16), (23), (16)(23)} in H_{6}?

My workings as below,

Since H_{6}, 6!= 1x2x3x4x5x6=720

Since H has 4 elements, i take 720/4=180 (Answer)

Please correct me if i am wrong.

Thanks.

Re: Find the no. of distinct left cosets of H = { e, (16), (23), (16)(23)} in H6.

By H6, I assume you mean the symmetric group on 6 elements. I've seen that denoted $\displaystyle S_6$ or $\displaystyle Sym\{1, 2,...,6\}$, but never $\displaystyle H_6$.

With that assumption, your reasoning and answer are correct.