M-groups and the supersolvable residual

Abstract

A finite group G is an M-group (monomial group) if each irreducible complex character of G is induced from a linear character of a subgroup of G. It is well known that M-groups are always solvable and that certain types of solvable groups are M-groups. In [5] Dornhoff proved a theorem implying that the class of M-groups is rather large. Recently Dade proved that each finite solvable group can be embedded in an M-group. In this paper we are concerned with the group theoretical structure of an M-group. In view of Dade's theorem this is clearly a difficult problem, although some theorems can be proved. If G is a solvable group we let G ~ be the supersolvable residual of G (the unique smallest normal subgroup of G having supersolvable quotient group). In [15] it was shown that for various types of M-groups G, the structure of G ~ is quite restricted. Moreover, the restrictions on G ~ are often much more severe than the restrictions on G ~ for various other saturated formations Y. To determine the structure of a particular M-group it was usually best to determine first the structure of its supersolvable residual. In view of these facts it seemed reasonable to investigate the supersolvable residual of a general M-group. For a solvable group G, the structure of G ~ is not sufficient to determine whether or not G is an M-group (see 3.7 and 3.8) although much information can be obtained by restricting attention to this subgroup. For example, it is shown that if G is an M-group, then the 2-Sylow subgroups of G S cannot be quaternion or the direct product of two quaternion groups. Restrictions on the odd Sylow subgroups of G ~ are also obtained. The paper is organized as follows. In Section 1 we set down the necessary preliminaries, state results from the theory of formations and give a result from character theory which can be sharpened when one is dealing with M-groups. In Section 2 we give a discussion of the connection between Mgroups and solvable groups. In this section most results from the literature on M-groups are generalized or improved. Included is a result showing that each finite solvable group can be embedded in an M-group having the same derived length. Section 3 contains the general restrictions on G s~ when G is an M-group. In some sense, this section contains an analysis of what happens (and what can't happen) in G s~ when G is an M-group, although G does not satisfy the hypotheses of the various theorems giving sufficient conditions for a group to be an M-group.