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 $k\geq 5$. Show that the index $[A_{k}:G]\geq k$.

(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.)

Title Simple Groups SimpleGroups 2013-03-22 19:30:43 2013-03-22 19:30:43 jac (4316) jac (4316) 6 jac (4316) msc 20B05