characteristic subgroup

If (G,*) is a group, then H is a characteristic subgroup of G (written HcharG) if every automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of G maps H to itself. That is, if fAut(G) and hH then f(h)H.

A few properties of characteristic subgroups:

Proofs of these properties:

  • Consider HcharG under the inner automorphismsMathworldPlanetmath of G. Since every automorphism preserves H, in particular every inner automorphism preserves H, and therefore g*h*g-1H for any gG and hH. This is precisely the definition of a normal subgroup.

  • Suppose H is the only subgroup of G of order n. In general, homomorphismsPlanetmathPlanetmathPlanetmathPlanetmath ( take subgroups to subgroups, and of course isomorphisms take subgroups to subgroups of the same order. But since there is only one subgroup of G of order n, any automorphism must take H to H, and so HcharG.

  • Take KcharH and HG, and consider the inner automorphisms of G (automorphisms of the form hg*h*g-1 for some gG). These all preserve H, and so are automorphisms of H. But any automorphism of H preserves K, so for any gG and kK, g*k*g-1K.

  • Let KcharH and HcharG, and let ϕ be an automorphism of G. Since HcharG, ϕ[H]=H, so ϕH, the restrictionPlanetmathPlanetmathPlanetmathPlanetmath of ϕ to H is an automorphism of H. Since KcharH, so ϕH[K]=K. But ϕH is just a restriction of ϕ, so ϕ[K]=K. Hence KcharG.

Title characteristic subgroup
Canonical name CharacteristicSubgroup
Date of creation 2013-03-22 12:50:56
Last modified on 2013-03-22 12:50:56
Owner yark (2760)
Last modified by yark (2760)
Numerical id 13
Author yark (2760)
Entry type Definition
Classification msc 20A05
Related topic FullyInvariantSubgroup
Related topic NormalSubgroup
Related topic SubnormalSubgroup
Defines characteristic