Full invariance is finite direct power-closed: Difference between revisions
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Suppose <math>H</math> is a [[fully invariant subgroup]] of a [[group]] <math>G</math>. For any positive integer <math>n</math>, consider the [[external direct product]] of <math>G</math> with itself <math>n</math>, and denote this by <math>G^n</math>. Let <math>H^n</math> be the subgroup comprising those elements where all coordinates are from within <math>H</math>. Then, <math>H^n</math> is a fully invariant subgroup of <math>G^n</math>. | Suppose <math>H</math> is a [[fully invariant subgroup]] of a [[group]] <math>G</math>. For any positive integer <math>n</math>, consider the [[external direct product]] of <math>G</math> with itself <math>n</math>, and denote this by <math>G^n</math>. Let <math>H^n</math> be the subgroup comprising those elements where all coordinates are from within <math>H</math>. Then, <math>H^n</math> is a fully invariant subgroup of <math>G^n</math>. | ||
==Related facts== | |||
===Opposite facts=== | |||
* [[Full invariance is not direct power-closed]] | |||
* [[Characteristicity is not direct power-closed]] | |||
===Similar facts=== | |||
* [[Homomorph-containment is finite direct power-closed]] | |||
* [[Normality-preserving endomorphism-invariance is finite direct power-closed]] | |||
* [[Verbality is finite direct power-closed]] | |||
Revision as of 20:35, 12 August 2010
This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) satisfying a subgroup metaproperty (i.e., finite direct power-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about fully invariant subgroup |Get facts that use property satisfaction of fully invariant subgroup | Get facts that use property satisfaction of fully invariant subgroup|Get more facts about finite direct power-closed subgroup property
Statement
Statement with symbols
Suppose is a fully invariant subgroup of a group . For any positive integer , consider the external direct product of with itself , and denote this by . Let be the subgroup comprising those elements where all coordinates are from within . Then, is a fully invariant subgroup of .
