Split and minimal abelian extensions of finite groups

Abstract:

Criteria for an abelian extension of a group to split are given in terms of a Sylow decomposition of the kernel and of normal series for the Sylow subgroups. An extension is minimal if only the entire extension is carried onto the given group by the canonical homomorphism. Various basic results on minimal extensions are given, and the structure question is related to the case of irreducible kernels of prime exponent. It is proved that an irreducible modular representation of ${\text {SL}}(2,p)$ or ${\text {PSL}}(2,p)$ for $p$ prime and $\geq 5$ afford a minimal extension with kernel of exponent $p$ only when the representation has degree 3, i.e., when the kernel has order ${p^3}$.