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characteristic subgroup
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(Definition)
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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:
Proofs of these properties:
- 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.
- Suppose
is the only subgroup of of order . In general, homomorphisms 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
.
- 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 ,
.
- 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
.
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"characteristic subgroup" is owned by yark. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: restriction, isomorphisms, order, preserves, inner automorphisms, normality of subgroups is not transitive, cardinality, subgroup, normal subgroup, maps, automorphism, group
There are 48 references to this entry.
This is version 10 of characteristic subgroup, born on 2002-07-21, modified 2006-02-17.
Object id is 3180, canonical name is CharacteristicSubgroup.
Accessed 8494 times total.
Classification:
| AMS MSC: | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) |
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Pending Errata and Addenda
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