Centrally indecomposable group
From Groupprops
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
View a complete list of group properties
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 satisfies the following equivalent conditions:
- It cannot be expressed as the central product of two proper subgroups.
- Every proper nontrivial central factor is a central subgroup, i.e., it is contained in the center of the group.
Definition with symbols
A group is said to be a centrally indecomposable group if it satisfies the following equivalent conditions:
- We cannot write
, viz., as a central product for proper subgroups
and
of
.
- Every proper nontrivial central factor of
is a central subgroup of
, i.e., it is contained in the center
.