Centrally indecomposable group: Difference between revisions
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A nontrivial [[group]] is said to be '''centrally indecomposable''' if it cannot be expressed as the central product of two proper subgroups. | 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. | Note that, for a [[centerless group]], this is equivalent to saying that there is no proper nontrivial [[central factor]]. However, for an group with a nontrivial center, the center itself is a central factor. | ||
===Definition with symbols=== | ===Definition with symbols=== | ||
Revision as of 01:47, 13 September 2009
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of simple group|Find other variations of simple group |
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 proper 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 .