Difference between revisions of "Congruent group extensions"

From Groupprops
Jump to: navigation, search
 
Line 16: Line 16:
 
1 \to & A \to & G_2 \to & B \to & 1 \\
 
1 \to & A \to & G_2 \to & B \to & 1 \\
 
\end{array}</math>
 
\end{array}</math>
 +
 +
==Related notions==
 +
 +
* [[Pseudo-congruent group extensions]]: This is a coarser equivalence relation, where we allow the maps on <math>A</math> and <math>B</math> to be [[automorphism]]s instead of requiring them to both be the identity map.

Latest revision as of 04:11, 3 May 2013

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 \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^{\operatorname{id}_A} & \downarrow^{\varphi} & \downarrow^{\operatorname{id}_B} & \downarrow\\
1 \to & A \to & G_2 \to & B \to & 1 \\
\end{array}

Related notions