characteristic subgroup
If (G,*) is a group, then H is a characteristic subgroup of G (written HcharG) if every automorphism of G maps H to itself. That is, if f∈Aut(G) and h∈H then f(h)∈H.
A few properties of characteristic subgroups:
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If HcharG then H is a normal subgroup
of G.
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If G has only one subgroup
of a given cardinality then that subgroup is characteristic.
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If KcharH and H⊴G then K⊴G. (Contrast with normality of subgroups is not transitive.)
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If KcharH and HcharG then KcharG.
Proofs of these properties:
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Consider HcharG under the inner automorphisms
of G. Since every automorphism preserves H, in particular every inner automorphism preserves H, and therefore g*h*g-1∈H for any g∈G and h∈H. This is precisely the definition of a normal subgroup.
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Suppose H is the only subgroup of G of order n. In general, homomorphisms
(http://planetmath.org/GroupHomomorphism) 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.
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Take KcharH and H⊴G, and consider the inner automorphisms of G (automorphisms of the form h↦g*h*g-1 for some g∈G). These all preserve H, and so are automorphisms of H. But any automorphism of H preserves K, so for any g∈G and k∈K, g*k*g-1∈K.
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Let KcharH and HcharG, and let ϕ be an automorphism of G. Since HcharG, ϕ[H]=H, so ϕH, the restriction
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 |