# 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 |