Split extension of groups and semidirect product

Question

I am studing Ext functor and have some basic problem.

For every semidirect product $G$ of groups $N$ and $H$, short exact sequence $0 \to N \to G \to H \to 0$ splits.

On the other hand, every short exact sequence which splits is a neutral element in Baer sum.

This means that for every $G$ which is semidirect product, sequence $0 \to N \to G \to H \to 0$ is equivalent to sequence $0 \to N \to N\times H\to H \to 0$.

In particular, $G \cong N\times H$, which of course don't have to be true (we can take abelian groups $N$ and $H$ and $G$ don't have to be abelian).

Where is the mistake in my reasoning?

Answer 1

Edit: I read your post carefully, and it looks like you mention the Baer sum at the very beginning. So let me try to explain two different things.

1. The Yoneda $\operatorname{Ext}^1$ is defined as the equivalence classes of extensions. Two extensions $$\tag{*} 0 \to B\to C\to A \to 0$$ and $$\tag{**} 0 \to B\to C'\to A \to 0$$ are equivalent if there exists a morphism $\phi\colon C\to C'$ that makes part of a commutative diagram with exact lines (*) and (**) and identity morphisms $id\colon B\to B$ and $id\colon A\to A$. Note that by the five-lemma, such a morphism $\phi$ is necessarily an isomorphism (and this observation shows that this is indeed a symmetric relation, while it is clearly reflexive and transitive).

   Since semi-direct product is not usually isomorphic to the direct product, the arrow $\phi$ simply can't exist and you don't get isomorphic extensions.

   Note that short exact sequences and the five-lemma still make sense for non-abelian groups (so you can take the same definition for the equivalence of extensions)... but the Baer sum doesn't!

2. Baer sum for exact sequences in an abelian category is defined as a certain construction on short exact sequences, and then one verifies that the construction does not depend on the equivalence relation. The zero element is the equivalence class of the extension $0 \to B\to B\oplus A\to A\to 0$. A good reference where the Yoneda's $\operatorname{Ext}^1$ with the Baer sum is constructed is Mac Lane's "Homology". The assumption that your category is abelian (e.g. that your exact sequences are formed by abelian groups) is crucial for the construction.