property of infinite simple group

Although much recent work has been done to classify the finite simple groups, infiniteMathworldPlanetmath simple groupsMathworldPlanetmathPlanetmath 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 subgroupsMathworldPlanetmath of finite index.


Let G be an infinite simple group and HG. Then G acts on the cosets of H and this induces a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath form G to Sn where n=[G:H]. If H has finite index in G then n is finite so G maps homomorphically into the finite groupMathworldPlanetmath Sn. 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 setsMathworldPlanetmath so we cannot invoke clever arguments about the configurationPlanetmathPlanetmath 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 𝔰𝔩2(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