Centrally indecomposable group: Difference between revisions
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viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>. | viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 01:13, 13 September 2009
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition
Symbol-free definition
A nontrivial group is said to be centrally indecomposable if it cannot be expressed as the central product of two proper subgroups.
Note that, for a centerless group, this is equivalent to saying that there is no nontrivial central factor. However, for an group with a nontrivial center, the center itself is a central factor.
Definition with symbols
A group is said to be a centrally indecomposable group if we cannot write:
viz., as a central product for proper subgroups and of .