Marcel Herzog

On Supersolvable Groups and the Nilpotator

Abstract A finite group G is called G a 𝒯-group if each subnormal subgroup of G is normal in G and a subgroup K of G is called an ℋ-subgroup of G if N G (K) ∩ K g ⊆ K for all g ∈ G. Using the notion of ℋ-subgroups, we present some new conditions for supersolvability and we characterize supersolvable groups, which are either 𝒯-groups or A-groups (i.e., all their Sylow subgroups are abelian). For example, we prove that if all cyclic subgroups of G of prime order or of order 4 are ℋ- subgroups of G, then G is supersolvable with a well defined structure. We also show, that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic ℋ-subgroups of G.

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveCrd=G, wherer=rankG.

Finite groups in which the degrees of the nonlinear irreducible characters are distinct

Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others. The purpose of this article is to consider the other extreme, namely, to characterize all finite groups in which all the nonlinear irreducible characters have distinct degrees. Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others (see [8, Chapter 12]). The purpose of this article is to consider the other extreme, namely, the case when all nonlinear irreducible character degrees are distinct. We prove Theorem. Let G be a nonabelian finite group. Let {0I, 02, . . , Or } be the set of all nonlinear irreducible ordinary characters of G. Assume that Oi (1)

Powers of Characters of Finite Groups

O (1) for all i 0 j . Then one of the following holds: (1) G is an extra-special 2-group. (2) G is a Frobenius group of order pnf(pn 1) for some prime power pn with an abelian Frobenius kernel of order pn and a cyclic Frobenius complement. (3) G is a Frobenius group of order 72 in which the Frobenius complement is isomorphic to the quaternion group of order 8. Remarks. All the groups described above satisfy the assumption of the theorem. The groups of type (1) and (2) have exactly one nonlinear character degree, while the group in (3) has two such character degrees. To show that there are no perfect groups satisfying the assumption, we use the classification of the finite simple groups. The proof for nonperfect groups is independent of the classification. In [10], Seitz shows that if in the theorem r = 1 , then either (1) or (2) holds. Notation. Most of our notation is standard and taken mainly from [8]. We will denote the set of all irreducible ordinary characters of the finite group G by Received by the editors September 7, 1990, and, in revised form, February 1, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 20CXX. This paper was written during the summer of 1990, when the second and third authors were visiting the University of Mainz, a visit supported by the DFG as part of the Darstellungstheorie project. We would like to thank the DFG for its support and the University of Mainz for its hospitality. ? 1992 American Mathematical Society 0002-9939/92

SMALL DOUBLING IN ORDERED GROUPS

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An extremal problem on the set of noncoprime divisors of a number

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Finite groups whose noncentral commuting elements have centralizers of equal size.

LetG be a finite group and θ a complex character ofG. Define Irr(θ) to be the set of all irreducible constituents of θ andIrr(G) to be the set of all irreducible characters ofG. Thecharacter-covering number of a finite groupG, ccn(G), is defined as the smallest positive integer m such thatIrr(χm) =Irr(G) for allχ∈Irr(G)—{1G}. If no such positive integer exists we say that the character-covering-number ofG is infinite. In this article we show that a finite nontrivial groupG has a finite character-covering-number if and only ifG is simple and non-abelian and ifG is a nonabelian simple group thenccn(G) ⩽ k2 − 3k + 4, wherek is the number of conjugacy classes ofG. Then we show (using the classification of the finite simple groups) that the only finite group with a character-covering-number equal to two is the smallest Jankos group,J1. These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.

On a combinatorial problem in group theory

We prove that if

The Br property and chromatic numbers of generalized graphs

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Direct and inverse problems in additive number theory and in non-abelian group theory

is a finite subset of an ordered group that generates a nonabelian ordered group, then