Difference between revisions of "Centrally indecomposable group"
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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 subcommunity that dabbles in such things) but is not very basic or common for people outside.
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Definition
Symbolfree definition
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 .
Formalisms
In terms of the simple group operator
This property is obtained by applying the simple group operator to the property: central factor
View other properties obtained by applying the simple group operator
The group property of being centrally indecomposable is obtained by applying the simple group operator to the subgroup property of being a central factor.