Pseudo-congruent group extensions

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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 \alpha \in \operatorname{Aut}(A), \beta \in \operatorname{Aut}(B), and 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^{\alpha} & \downarrow^{\varphi} & \downarrow^{\beta} & \downarrow\\
1 \to & A \to & G_2 \to & B \to & 1 \\
\end{array}

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.