Give a composition series for $S_4, S_5$ & $S_{n \ge 5}$, explaining why it is a composition series. Also give the composition factors for $S_4, S_5$.

I have been given the answers for the composition series for $S_4, S_5$ and their factors but I don't know how this result was achieved.

Please can someone give me a hint and then I can apply that also to $S_{n \ge 5}$.

The answers that I have been given are:

For $S_4$:

$G_1 = A_4$; $G_2 = \langle (12)(34), (13)(24) \rangle$; $G_3 = \langle (12)(34) \rangle$; $G_4 = 1$

Composition factors: $G|G_1 \cong \mathbb{Z}_2$; $G_1|G_2 \cong \mathbb{Z}_3$; $G_2|G_3 \cong \mathbb{Z}_2 \cong G_3|G_4$

For $S_5$:

$G_1 = A_5$; $G_2 = 1$

Composition factors: $G|G_1 \cong \mathbb{Z}_2$; $G_1|G_2 \cong A_5$

I don't understand how the results were obtained and I prefer to be able to work out the result, rather than just try and remember it. Thank you for any help!

  • $\begingroup$ Do you know that $A_n$ is simple for $n \ge 5$? $\endgroup$ – Max Jan 22 '16 at 1:00
  • $\begingroup$ @Max I knew that $A_5$ was simple. Does that mean that for $S_{n \ge 5}$, $G_1 = A_n$ and $G_2 = 1$? And if so, how do I work out the composition factors? $\endgroup$ – Dom Jan 22 '16 at 1:08

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