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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".

Eric M Downes (Jun 13 2024 at 13:01):

I don't know if you're still looking, but this came up recently in my literature search!

It doesn't directly answer what you want, but its close. Tim van der Linden has developed (co)homology for semi-abelian categories, starting with categories that have a zero-object. Here is his book.

He explicitly develops it at a very nice basic level for non-associative algebra in this preprint. I mention non-associativity because sometimes $G/H$ for non-normal $H$ is a "loop" (not-associative, has left and right division, and a two-sided identity)... not always, though.... this is due to either Baer or Brück IIRC, but is not a uniform method and was never developed systematically that I have found. I can find the paper if you want.