Difference between revisions of "Centrally indecomposable group"
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Revision as of 23:14, 7 May 2008
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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A group is said to be centrally indecomposable if it satisfies the following equivalent conditions:
- It has no proper nontrivial central factor
- It cannot be expressed as the central product of two proper subgroups, or equivalently, for any proper subgroup, the product with its centralizer is again proper.
Definition with symbols
A group is said to be a centrally indecomposable group if we cannot write:
viz, a central product for nontrivial groups and .
In terms of the simple group operator
This property is obtained by applying the simple group operator to the property: central factor
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