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 $k\\geq 5$ . 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.)

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单词	SimpleGroups
释义	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 $k\\geq 5$ . 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.) …
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