Discussion
Consider the following statements :
(I) Every finite group is isomorphic to a subgroup of the alternating group An for some \(n \gt 1\).
(II) The symmetric group Sn is nilpotent for \(n \ge 3\).
Which is correct ?
Answer: A
Only (I) is correct.
(I) By Cayley's theorem a finite group G embeds in \(S_{m}\) with \(m = |G|\). The map sending a permutation \(\sigma\) of {1, ..., m} to \(\sigma\) itself when \(\sigma\) is even, and to \(\sigma\) times the transposition \((m{+}1\ m{+}2)\) when \(\sigma\) is odd, embeds \(S_{m}\) in \(A_{m+2}\). So G is isomorphic to a subgroup of some \(A_{n}\).
(II) \(S_{n}\) is not nilpotent for \(n \ge 3\). A nilpotent group has a non-trivial centre, but the centre of \(S_{n}\) is trivial for \(n \ge 3\); for \(S_{3}\), which has order 6, nilpotency would also make it the product of its Sylow subgroups and hence abelian, which it is not.
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