Simple Groups

Recall that a group $G$ is simple if it has no normalsubgroups except itself and $\{e\}$. Let $G$ be a finitesimple group and let $p$ be a prime number.

(a) Suppose $G$ has precisely $k$ Sylow $p$-subgroups with$k>1$. Show that $G$ is isomorphic to a subgroup of thesymmetric group $S_k$.

(b) With the same hypothesis, show that $G$ is isomorphicto a subgroup of the alternating group $A_k$.

(c) Suppose $G$ is a simple group that is a propersubgroup of $A_k$ and $k\ge 5$. Show that the index$[A_k:G]\ge k$.

(d) Prove that if $G$ is a group of order $120$ then $G$is not a simple group. (Parts (b) and (c) may be helpful.)

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