# Pseudo-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. Explicitly, this means that there are automorphisms , , and an isomorphism such that the following diagram commutes:

## Related notions

- Congruent group extensions: This is a finer equivalence relation imposed on group extensions, where we require the automorphisms of and to both be identity maps.