NORMAL PI-COMPLEMENT THEOREMS

It is a well-known theorem of Frobenius that a finite group G has a no rmal p-complement if and only if two elements of its Sylow p-subgroup that are conjugate in G are already conjugate in P. This result was ge neralized by Brauer and Suzuki, see e.g. [2], from Sylow to Hall subgr oups using additional conditions, namely if N is a Hall pi-subgroup of G and two elements of H that are conjugate in G are already conjugate in N and each elementary pi-subgroup in G can be conjugated into H th en G has a normal pi-complement. In this paper we generalize the theor em of Frobenius from Sylow to Hall subgroups under different condition s, the conjugacy condition is restricted only for elements of odd prim e order and elements of order 2 and 4 in N, on the other hand we assum e that H has a Sylow tower. This also generalizes a result of Zappa, s ee [8], saying that if H is a Hall pi-subgroup of G with a Sylow tower , and two elements of H that are conjugate in G are already conjugate in H, then G has a normal pi-complement. As a corollary we get a weake ning of the conditions of another result of Zappa, saying that if a fi nite group has a Hall-pi-subgroup H with a Sylow tower and N possesses a set of complete right coset representatives, which is invariant und er conjugation by H, then G has a normal pi-complement. In the end we generalize the theorem of Brauer and Suzuki in another direction, name ly assuming that G has a solvable Hall pi-subgroup and every elementar y pi-subgroup of G can be conjugated into it, and if two elements of N of prime power order in H that are conjugate in G are already conjuga te in N, then G has a normal pi-complement. In this paper all groups a re finite. For basic definitions the reader is referred to [6].