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Peter Schmid
August 31, 2009

### Abstract

Let N be a normal subgroup of the finite group X and let G = X / N . It is well known that the number k ( X ) = |Irr( X )| of conjugacy classes (irreducible characters) of X is bounded above by the product k ( N ) k ( G ). For more precise results one has to determine, for any θ ∈ Irr( N ), the number of G -conjugates of θ and the number k θ ( G ) of irreducible characters of X lying above θ . Clifford theory reduces the computation of k θ ( G ) to the situation where N = Z is central in X , and then it only depends on the Clifford obstruction μ = μ G ( θ ). The class number k μ ( G ) is defined for any μ ∈ H 2 ( G , ℂ*), and behaves well when passing to isoclinic central extensions. Suppose that μ ≠ 1. There are examples where k μ ( G ) = 1, in which case X is a group of central type, and examples where k μ ( G ) = k ( G ) is as large as possible. In the first case G must be solvable (Howlett–Isaacs). The latter case can happen when G is perfect but not when G is a simple or, more generally, a quasisimple group.

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Gunter Malle
August 31, 2009

### Abstract

We show that for each finite non-abelian simple group and each prime p either there exists an irreducible Brauer character which takes the value zero on some p -regular element, or p = 2 and all degrees of irreducible Brauer characters are powers of 2. This generalizes an old result of Burnside from ordinary to Brauer characters. For the proof we introduce the notion of the defect zero graph of a finite group, which may be of independent interest.

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Silvio Dolfi, Emanuele Pacifici, Lucia Sanus, Pablo Spiga
August 31, 2009

### Abstract

Let G be a finite group, and Irr( G ) the set of irreducible complex characters of G . We say that an element g ∈ G is a vanishing element of G if there exists χ in Irr( G ) such that χ ( g ) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G , and we define the prime graph on it, which we denote by Γ( G ). Focusing on the class of solvable groups, we prove that Γ( G ) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Γ( G ) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Γ( G ).

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Joachim König
August 31, 2009

### Abstract

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L -series suggest a generalization of these groups, namely to groups with the property that every irreducible character has some multiple which is induced from a character φ of U with solvable factor group U /ker( φ ). Using the classification of finite simple groups, we prove that these groups are also solvable. This means in particular that the mentioned properties do not enable one to deduce a proof of the famous Artin conjecture for any non-solvable group from a possible proof for solvable groups.

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A. Giambruno, C. Polcino Milies, Sudarshan K. Sehgal
August 31, 2009

### Abstract

Let ∗ be an involution of a group algebra FG induced by an involution of the group G . For char F ≠ 2, we classify the torsion groups G with no elements of order 2 whose Lie algebra of ∗-skew elements is nilpotent.

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Keith A. Kearnes
August 31, 2009

### Abstract

We show that any variety of groups that contains a finite nonsolvable group contains an axiomatic formation that is not a subvariety.

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Peter Hegarty, Desmond MacHale
July 30, 2009

### Abstract

We show that 3 7 is the smallest order of a non-trivial odd order group which occurs as the full automorphism group of a finite group.

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Zhencai Shen, Shirong Li, Wujie Shi
March 11, 2010

### Abstract

A subgroup H of the finite group G is said to be quasinormally (resp. S -quasinormally) embedded in G if for every Sylow subgroup P of H , there is a quasinormal (resp. S -quasinormal) subgroup K in G such that P is also a Sylow subgroup of K . Groups with certain quasinormally (resp. S -quasinormally) embedded subgroups of prime-power order are studied. For example, if a group G has a normal subgroup H such that G / H ∈ ℱ and such that for each Sylow subgroup P of H , every member in some ℳ d ( P ) is quasinormally embedded in G , then G ∈ ℱ: here ℳ d ( P ) is a set of maximal subgroups of P with intersection the Frattini subgroup.

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Xiuyun Guo, Xianbiao Wei
August 31, 2009

### Abstract

A subgroup H is said to be an ℋ-subgroup of a group G if H g ∩ N G ( H ) ⩽ H for all g ∈ G . In this paper, we investigate the structure of a group G under the assumption that every subgroup of order p m of a Sylow p -subgroup of G belongs to ℋ( G ) for a given positive integer m . Some results related to p -nilpotence and supersolvability of a group G are obtained.

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Daciberg Lima Gonçalves, John Guaschi
July 14, 2009

### Abstract

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups P n (ℝ P 2 ) of the projective plane. The maximal finite subgroups of P n (ℝ P 2 ) are isomorphic to the quaternion group of order 8 if n = 3, and to ℤ 4 if n ⩾ 4. Further, for all n ⩾ 3, the following groups are, up to isomorphism, the infinite virtually cyclic subgroups of P n (ℝ P 2 ): ℤ, ℤ 2 × ℤ and the amalgamated product ℤ 4 ∗ ℤ 2 ℤ 4 .

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Aleksander Ivanov
July 30, 2009

### Abstract

We embed a countably categorical group G into a locally compact group Ḡ with a non-trivial topology and study how topological properties of Ḡ are related to the structure of definable subgroups of G .