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Stream: theory: algebraic topology

Topic: non-normal group "extensions"

Mike Shulman (Feb 07 2022 at 01:40):

If $H$ is a normal subgroup of $G$, there is a short exact sequence $1 \to H \to G \to G/H \to 1$. And conversely, given $H$ and $G/H$, the possible such short exact sequences are classified by appropriate actions and group cohomology.

Is there any analogous classification in the case when $H\le G$ is not normal? Then $G/H$ is no longer a group, but we can still recover the underlying set of $G$ as isomorphic to $G/H \times H$; is there any vaguely cohomological way to encode the group structure of $G$ in terms of a subgroup $H$ and a set $G/H$?

John Baez (Feb 07 2022 at 02:36):

I haven't seen the thing you're looking for.

Mike Shulman (Feb 07 2022 at 05:17):

Darn, you were my best hope.

John Baez (Feb 07 2022 at 21:46):

Sorry, it's an interesting idea but I haven't seen "abnormal group cohomology".