Example of a non-splitting exact sequence $1 \to K \to G \to H \to 1$

Question

Suppose $G$, $K$ and $H$ are finitely generated infinite groups such that
$$1 \to K \to G \to H \to 1$$ is a short exact sequence.

Question: Are there examples where this sequence does not split?

There are examples of such groups when $H$ is finite(Does every short exact sequence split?) or when $G$ is not a finitely generated group (Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$).

Thank you for your help.

$1 \to \mathbb Z \times C_2 \to \mathbb Z^2 \times C_4 \to \mathbb Z \times C_2 \to 1$. — Carla only proves trivial prop, Oct 31 '22 at 13:17

Bill Cook · Answer 1 · 2022-10-31T18:28:27.023

4

$0 \to \mathbb{Z}^2 \to \mathbb{Z}^2 \to \mathbb{Z}\oplus\mathbb{Z}_2 \to 0$

using maps $(x,y) \mapsto (x,2y)$ and $(x,y) \to (x,y\text{ mod }2)$.

The only morphism from $\mathbb{Z}_2$ to $\mathbb{Z}$ is trivial, so you cannot split.

In more detail: Say $f(x,y) = (x,y\text{ mod }2)$ is our map from $\mathbb{Z}^2 \to \mathbb{Z}\oplus \mathbb{Z}_2$. Now $\mathbb{Z}^2$ doesn't have any elements of order 2, so to have a morphism $g:\mathbb{Z} \oplus \mathbb{Z}_2 \to \mathbb{Z}^2$, you are forced to send $g(0,1)$ to $(0,0)$. Thus $g \circ f$ cannot be the identity. [There's no split.]

edited Oct 31 '22 at 18:28

answered Oct 31 '22 at 13:20

Bill Cook

30,374

1

$\mathbb Z_2$ is not infinite. – Carla only proves trivial prop Oct 31 '22 at 13:23
To be fair, they ask for all three groups to be infinite. To fix that, you can just add an extra factor of $\mathbb{Z}$ to each group, and define the maps to be the identity on those components. – Steven Gubkin Oct 31 '22 at 13:23
I guess I didn't read the question very carefully. :P – Bill Cook Oct 31 '22 at 13:27
Thank you, I don't think $Im(f) = ker(g)$ in this case? – ghc1997 Oct 31 '22 at 15:22
The first group should be $\mathbb{Z}$ – Alvaro Martinez Oct 31 '22 at 16:20
Ok. Now this is fixed. I had accidentally flipped my factors around (but not consistently). – Bill Cook Oct 31 '22 at 18:31

Example of a non-splitting exact sequence $1 \to K \to G \to H \to 1$

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