Notes on Splitting Extensions of Groups

Tang, C.Y.

Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 371-374

Résumé

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup φ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition φ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂φ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) φ⊂φ(G). It follows that φ(G) ∩ N ≠ 1. Thus the condition φ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

Détail

DOI : 10.4153/CMB-1968-041-3