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 $\phi :G_1\to G_2$ such that the following diagram commutes:

$\begin{array}{ccccc}1\to & A\to & G_1\to & B\to & 1\\ \downarrow & {\downarrow }^{{\mathrm{i}}_{\mathrm{d}}}& {\downarrow }^{\phi }& {\downarrow }^{{\mathrm{i}}_{\mathrm{d}}}& \downarrow \\ 1\to & A\to & G_2\to & B\to & 1\\ \end{array}$

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.