Give an example of a split extension 0 →K →G p→ Q →1 in Groups for which there does not exist a

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Give an example of a split extension 0 →K →G p→ Q →1 in...

Give an example of a split extension $0 \rightarrow K \rightarrow G \stackrel{p}{\rightarrow} Q \rightarrow 1$ in Groups for which there does not exist a homomorphism $q: G \rightarrow K$ with $q i=1_{K}$. Compare with Exercise $2.8$.

Key Concepts

Section and Retraction

In the context of split extensions, a section is a group homomorphism from Q to G that provides a 'splitting' of the exact sequence by reversing the projection from G to Q. However, having a section does not imply that there exists a retraction (a homomorphism from G to K that is the identity on K). The absence of such a retract, even in a split extension, highlights the subtle differences between having a complementary subgroup via semidirect product decomposition and having a projection map onto the kernel that is a group homomorphism.

Semidirect Product

A semidirect product is a construction that arises naturally in the context of split extensions. When a group G is expressed as the semidirect product K ? Q, the subgroup K is normal in G and Q acts on K via conjugation determined by the splitting section. Unlike a direct product, this structure may incorporate nontrivial interaction between K and Q, which can prevent the existence of a homomorphism from G onto K that restricts to the identity on K.

Group Extension

A group extension is a way of constructing a new group G from a normal subgroup K and a quotient group Q, usually represented by a short exact sequence 0 ? K ? G ? Q ? 1. This sequence indicates that G contains a copy of K as a normal subgroup and that the quotient of G by K is isomorphic to Q. Understanding group extensions is fundamental to studying how more complex groups can be built from simpler pieces.

Split Extension

A split extension is a special type of group extension where there exists a section—a homomorphism from Q back to G—that acts as a right inverse to the projection map. In other words, the short exact sequence 0 ? K ? G ? Q ? 1 splits if there is a homomorphism s: Q ? G satisfying p s = 1_Q. This guarantees that G is isomorphic to the semidirect product K ? Q, although it does not imply the existence of a homomorphic retraction from G to K.

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