A subgroup H of a group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B such that G = HB and H permutes with every Sylow subgroup of B.
A group G [member of] F if and only if there is a normal subgroup H of G such that G/H [member of] F and every maximal subgroup of any Sylow subgroup of H is either E-supplemented or SS-quasinormald in G.
Next let C be a subgroup of order 2 that will remain fixed throughout the discussion, and let N(C) be its normalizer (= the unique Sylow subgroup
containing C); as noted in Section 1 the group N(C)/C acts by reflection on Fix (C, [Sigma]) [approximately equal to] [S.sup.1].
A subgroup H of a group G is said to be S-quasinormal (or [pi]-quasinormal) in G if H permutes with all Sylow subgroups of G, i.e., HS = SH for any Sylow subgroup S of G.
In this paper, we will try an attempt to unify the two concepts and establish the structure of groups under the assumption that all maximal subgroups or all minimal subgroups of a Sylow subgroup or are SCAP or S-supplemented subgroups.
Hall proved that a group G is soluble if and only if every Sylow subgroup
of G is complemented in G.
His topics include the Schur-Zassenhaus theorem: a bit of history and motivation, abelian and minimal normal subgroups, normal subgroups with abelian sylow subgroups
, and groups with specific classes of subgroups complemented.
In 1992, Bi (BI, 1992) showed that [L.sub.2] ([p.sup.k]) can be characterized only by the order of normalizer of its Sylow subgroups. In other words, if G is a group and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where P [member of] Sylr (G) and Q [member of] [Syl.sub.r] ([L.sub.2] ([p.sup.k])) for every prime r, then G [congruent to] [L.sub.2]([p.sup.k]).
Comparing this type of characterization with characterization by orders of normalizers of Sylow subgroups, it seems that characterization by the number of Sylow subgroups is much stronger than characterization by orders of normalizers of Sylow subgroups.
This notion can be strengthen in various ways, for example one can say that a subgroup H is S-permutable (or S-quasinormal) in G, if H permutes with all Sylow subgroups of G (for all primes in the set [pi](G) of the prime divisors of [absolute value G]).
For instance, [1, 2] describe the structure of the groups in which the subnormal subgroups permute with all Sylow subgroups (called PST-groups).
Under this condition, all Sylow subgroups
of G are abelian and so G is an M-group by Theorem 6.23 of .