Congruent group extensions

Definition

Suppose $A$ and $B$ are (possibly isomorphic, possibly non-isomorphic groups). Consider two group extensions $G_{1},G_{2}$ both "with normal subgroup $A$ and quotient group $B$." Explicitly, this means we are given two short exact sequences:

$1\to A\to G_{1}\to B\to 1$

and:

$1\to A\to G_{2}\to B\to 1$

We say that the group extensions are congruent if there is an isomorphism between the short exact sequences that restricts to the identity maps on $A$ and $B$ respectively. Explicitly, this means that there is an isomorphism $\varphi :G_{1}\to G_{2}$ such that the following diagram commutes:

${\begin{array}{lllll}1\to &A\to &G_{1}\to &B\to &1\\\downarrow &\downarrow ^{\operatorname {id} _{A}}&\downarrow ^{\varphi }&\downarrow ^{\operatorname {id} _{B}}&\downarrow \\1\to &A\to &G_{2}\to &B\to &1\\\end{array}}$

Related notions

• Pseudo-congruent group extensions: This is a coarser equivalence relation, where we allow the maps on $A$ and $B$ to be automorphisms instead of requiring them to both be the identity map.