Simple Groups

Recall that a group $G$ is simple if it has no normal subgroups except itself and $\{e\}$. Let $G$ be a finite simple 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 the symmetric group $S_k$.

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

(c) Suppose $G$ is a simple group that is a proper subgroup 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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Title	Simple Groups
Canonical name	SimpleGroups
Date of creation	2013-03-22 19:30:43
Last modified on	2013-03-22 19:30:43
Owner	jac (4316)
Last modified by	jac (4316)
Numerical id	6
Author	jac (4316)
Classification	msc 20B05