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.