A subgroup H is said to be nc-supplemented in a group G if there is a subgroup K ≤ G such that HK G and H ∩ K is contained in HG, the core of H in G. We characterize the solvability of finite groups G with some subgroups of Sylow subgroups nc-supplemented in G. We also give a result on c-supplemented subgroups.

The aim of this paper is to give a direct approach to the study of the Sylow ^-subgroups Sn of the symmetric group of degree pn. [We assume throughout that p^2.] Many of the results are already known and are treated in a paper by Kaloujnine where he uses a particular representation by means of "reduced polynomials."1 It has seemed worth while to restate some of his results using the concept of ...