Weak stem extension

Definition

Consider a group extension $G$ of $N$ and $Q$ as follows: $G$ is a group, $N$ is a normal subgroup of $G$, and $Q$ is the quotient group $G/N$.

We say that this group extension is a weak stem extension if the following hold:

1. The extension is a central extension, i.e., $N$ is a normal subgroup of $G$.
2. The associated homomorphism $N\otimes N\to G^{\operatorname {ab} }\otimes N$ is the trivial homomorphism.

Relation with other properties

Stronger properties of group extensions

• stem extension

Weaker properties of group extensions

• central extension

References

• On the homology theory of central group extensions: I—The commutator map and stem extensions by Beno Eckmann, Peter J. Hilton and Urs Stammbach, Commentarii Mathematici Helvetici, Volume 47,Number 1, Page 102 - 122(Year 1972): ^{Official copy (gated)More info}, definition mentioned after expression (1.6) on the second page of the paper.