Direct factor over central subgroup

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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