# Congruent group extensions

From Groupprops

## Definition

Suppose and are (possibly isomorphic, possibly non-isomorphic groups). Consider two group extensions both "with normal subgroup and quotient group ." Explicitly, this means we are given two short exact sequences:

and:

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 and respectively. Explicitly, this means that there is an isomorphism such that the following diagram commutes:

## Related notions

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