Pseudo-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. Explicitly, this means that there are automorphisms $α \in A u t (A)$ , $β \in A u t (B)$ , and an isomorphism $φ : G_{1} \to G_{2}$ such that the following diagram commutes:

$\begin{matrix} 1 \to & A \to & G_{1} \to & B \to & 1 \\ ↓ & ↓^{α} & ↓^{φ} & ↓^{β} & ↓ \\ 1 \to & A \to & G_{2} \to & B \to & 1 \end{matrix}$

Related notions

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