# property of infinite simple group

Although much recent work has been done to classify the finite simple groups, infinite^{} simple groups^{} have properties which make the study more difficult. Among them is the following basic result.

###### Theorem 1.

If a group is infinite and simple then it has no proper subgroups^{} of finite index.

###### Proof.

Let $G$ be an infinite simple group and $H\le G$. Then $G$ acts on the cosets of $H$ and this induces a homomorphism^{} form $G$ to ${S}_{n}$ where $n=[G:H]$. If $H$ has finite index in $G$ then $n$ is finite so $G$ maps homomorphically into the finite group^{} ${S}_{n}$. Thus the kernel of the homomorphism is non-trivial. As $G$ is simple, the kernel is $G$. As $H$ contains the kernel, $H=G$.
∎

This means that infinite simple groups do not act on finite sets^{} so we cannot invoke clever arguments about the configuration^{} of numbers. However linear representations may still apply. For example, $PSL(2,k)$ for an infinite field $k$ is simple, infinite, and can be represented in $SL(3,k)$ through the exponential map of a Chevalley basis of the Lie algebra $\U0001d530{\U0001d529}_{2}(k)$.

Title | property of infinite simple group |
---|---|

Canonical name | PropertyOfInfiniteSimpleGroup |

Date of creation | 2013-03-22 16:08:21 |

Last modified on | 2013-03-22 16:08:21 |

Owner | Algeboy (12884) |

Last modified by | Algeboy (12884) |

Numerical id | 5 |

Author | Algeboy (12884) |

Entry type | Result |

Classification | msc 20E32 |

Related topic | ExistenceOfMaximalSubgroups |