6533b7dcfe1ef96bd12732c5

RESEARCH PRODUCT

Primitive characters of subgroups ofM-groups

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One of the hardest areas in the Character Theory of Solvable Groups continues to be the monomial groups. A finite group is said to be an M-group (or monomial) if all of its irreducible characters are monomial, that is to say, induced from linear characters. Two are still the main problems on M-groups: are Hall subgroups of M groups monomial? Under certain oddness hypothesis, are normal subgroups of M-groups monomial? In both cases there is evidence that this could be the case: the primitive characters of the subgroups in question are the linear characters. This is the best result up to date ([4], [6]). Recently, some idea appears to be taking form. In [14], T. Okuyama proved that if G is an M-group and P is a Sylow p-subgroup of G, then N~;(P)/P is an M-group. Some time later, and more generally, I. M. lsaacs shows that if H is a Hall subgroup of an M-group then N s ( H ) / H ~ is also an M-group ([91). Since it is well known that normal Hall subgroups of M-groups are M-groups (in this case the strong hypothesis of normality makes the problem immediate), perhaps it is not only true that Hall subgroups of M-groups are M-groups but something stronger: that Hall subgroup normalizers also are. In this direction, we prove. Theorem A, Let G be an M-group and let H be a Hall subgroup o f G. I f c~ E I r r (Nc(H)) is primitive then c~ is linear.