characteristic subgroup
If is a group, then is a characteristic subgroup of (written ) if every automorphism of maps to itself. That is, if and then .
A few properties of characteristic subgroups:
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If then is a normal subgroup of .
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If has only one subgroup of a given cardinality then that subgroup is characteristic.
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If and then . (Contrast with normality of subgroups is not transitive.)
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If and then .
Proofs of these properties:
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Consider under the inner automorphisms of . Since every automorphism preserves , in particular every inner automorphism preserves , and therefore for any and . This is precisely the definition of a normal subgroup.
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Suppose is the only subgroup of of order . 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 of order , any automorphism must take to , and so .
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Take and , and consider the inner automorphisms of (automorphisms of the form for some ). These all preserve , and so are automorphisms of . But any automorphism of preserves , so for any and , .
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Let and , and let be an automorphism of . Since , , so , the restriction of to is an automorphism of . Since , so . But is just a restriction of , so . Hence .
Title | characteristic subgroup |
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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 |