Direct factor over central subgroup

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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

Suppose H is a subgroup of a group G. We say that H is a direct factor over central subgroup of G if it satisfies the following equivalent conditions:

  1. There exists a subgroup A of H such that A is a central subgroup of G and H/A is a direct factor of the quotient group G/A.
  2. In the quotient map \rho:G \to G/Z(G), where Z(G) is the center of G, \rho(H) is a direct factor of \rho(G) = G/Z(G).

Relation with other properties

Stronger properties

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

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Normal subgroup