It was easy to find the composition factors of $S_3 \times Z_2$. However, I cannot see how to find 'all' possible composition series of this group. It seems like a formidable work to me... Could anyone please help?
2026-03-26 04:33:59.1774499639
Finding all composition series of $S_3 \times Z_2$
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By Jordan-Hölder, composition series are unique up to permutation of composition factors, which in this case are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, and $\mathbb{Z}/3\mathbb{Z}$. Suppose $1\lhd G\lhd H\lhd S_3\times \mathbb{Z}/2\mathbb{Z}$ is a composition series, then $H$ must have order 4 or 6.
Exercise: There are no normal subgroups of order 4
Finally, we look at subgroups of order 6. Note that every subgroup of order 6 has index 2, hence is automatically normal. There are exactly 3 subgroups of order 6, 2 of which are ismorphic to $S_3$, and one of which is isomorphic to $\mathbb{Z}/6\mathbb{Z}$. Note that $S_3$ has a unique composition series $1\lhd A_3 \lhd S_3$, and $\mathbb{Z}/6\mathbb{Z}$ has 2 composition series. It follows that there are 4 different composition series in total, which I leave for you to write down.