Unfactorizable 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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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 simplicity|Find other variations of simplicity | Read a survey article on varying simplicity


Symbol-free definition

A group is said to be unfactorizable if it satisfies the following equivalent conditions:

  • It has no proper nontrivial permutably complemented subgroup
  • It does not possess an exact factorization (viz it cannot be written as the product of a matched pair of subgroups)

Definition with symbols

A group G is said to be unfactorizable if for any choice of subgroups H and K such that H \cap K is trivial and HK= G, H and K are (in some order) the whole group and the trivial subgroup.

In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: permutably complemented subgroup
View other properties obtained by applying the simple group operator

Relation with other properties

Weaker properties