Join of direct factor and central subgroup

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

Definition with symbols

A subgroup H of a group G is termed a join of direct factor and central subgroup or a central subgroup over direct factor if it satisfies the following equivalent conditions:

  1. There exist subgroups A,B of G such that A is a direct factor of G, B is a central subgroup of G, and H is the join (in this case, also the product) of A and B.
  2. There exists a direct factor A of G contained in H such that H/A is a central subgroup of the quotient group G/A.

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Direct factor obvious |FULL LIST, MORE INFO
Central subgroup obvious |FULL LIST, MORE INFO

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Central factor product with its centralizer is the whole group |FULL LIST, MORE INFO
Join-transitively central factor join with any central factor is a central factor |FULL LIST, MORE INFO
Direct factor over central subgroup image in quotient by center is a direct factor |FULL LIST, MORE INFO