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November 6, 2008
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In this article we show that the isomorphism type of certain semilinear classical groups may depend on the choice of form matrix, as well as the dimension and the field size. When there is more than one isomorphism type, we count them and present effective polynomial-time algorithms to determine whether two such groups are isomorphic.

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November 6, 2008
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In this paper we determine all algebraic transformation groups G , defined over an algebraically closed field k, which operate transitively, but not primitively, on a variety Ω, subject to the following conditions. We require that the (non-effective) action of G on the variety of blocks is sharply 2-transitive, as well as the action on a block Δ of the normalizer G Δ . Also we require sharp transitivity on pairs ( X, Y ) of independent points of Ω, i.e. points contained in different blocks.

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November 6, 2008
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If π is a set of primes, a finite group G is called block π -separated if for every two distinct irreducible complex characters α, β ∈ Irr( G ) there is a prime p ∈ π such that α and β are in different p -blocks. The group G is called principally π -separated if the above holds whenever β = 1 G . Bessenrodt and Zhang conjectured that if G is a solvable principally π -separated group then G is π -separated. We construct a family of counter-examples to this conjecture.

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November 6, 2008
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Let p be an odd prime and let k be an algebraically closed field of characteristic p . Also, let G be a finite p ′-group. By Maschke's theorem, kG is isomorphic to a product End k ( V i ) as a k -algebra. Suppose that a p -subgroup P of Aut( G ) stabilizes End k ( V i 0 ) for some i 0 . Such a V i 0 will be an endo-permutation kP -module. Puig showed that the only modules that occur in this way are those whose image is torsion in the Dade group D ( P ). If G is any finite group and b is a defect zero block of kG , then kGb ≅ End k ( L ) for some L . If kGb is P -stable for some p -subgroup P of Aut( G ) and Br P ( b ) ≠ 0, then L will again be an endo-permutation kP -module. We show that if p ⩾ 5, then L is torsion in D ( P ). This result depends on the classification of the finite simple groups.

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November 6, 2008
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We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups.

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November 6, 2008
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Let p be a prime. The goal of this paper is to classify the finite groups with exactly one conjugacy class of size a multiple of p .

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November 6, 2008
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Let G be a connected reductive algebraic group defined over 𝔽 q , where q is a power of a prime p that is good for G . Let F be the Frobenius morphism associated with the 𝔽 q -structure on G and set G = G F , the fixed point subgroup of F . Let P be an F -stable parabolic subgroup of G and let U be the unipotent radical of P ; set P = P F and U = U F . Let G uni be the set of unipotent elements in G . In this note we show that the number of conjugacy classes of U in G uni is given by a polynomial in q with integer coefficients.

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November 6, 2008
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We generalize the definition of the precise center of a group to the crossed modules context. We construct the Ganea map for the homology of crossed modules, and we study the connections between the precise center of a crossed module and the Ganea map. We extend some other known notions from group theory such as capable and relatively capable groups, capable pairs and unicentral groups with the definitions of capable and unicentral crossed modules. Finally we show how to apply these constructions to solve some open questions in the theory of crossed modules.

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November 6, 2008
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A subgroup H is called ℳ -supplemented in a finite group G if there exists a subgroup B of G such that G = HB and such that H 1 B is a proper subgroup of G for any maximal subgroup H 1 of H . In this paper we fix a subgroup D in every non-cyclic Sylow subgroup P of G satisfying 1 < D < P and study the structure of G under the assumption that all subgroups H of P with | H | = | D | are ℳ -supplemented in G or have a supersolvable supplement in G .

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November 6, 2008
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We determine the non-abelian finite p -groups G with C G ( x ) ⩽ H for any minimal non-abelian subgroup H of G and each x ∈ H – Z ( G ) (Theorem 1.1). This solves Problem 757 of Berkovich [Groups of prime power order, vol. 1 and vol. 2, 2008]. We also classify the non-abelian finite p -groups G such that whenever A is a maximal subgroup of any minimal non-abelian subgroup H in G , then A is also a maximal abelian subgroup in G (Theorem 1.2), and this solves another problem of Berkovich [Groups of prime power order, vol. 1 and vol. 2, 2008]. Finally, we generalize a result of Blackburn [J. Algebra 3: 30–37, 1966] concerning finite p -groups in which the non-normal subgroups have non-trivial intersection (Theorem 1.3 and Corollary 1.4).

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November 6, 2008
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November 6, 2008
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Equations in free groups have become prominent recently in connection with the solution to the well-known Tarski conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P -space. The special case of one-variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial-time algorithm for finding solution sets. One-variable equations are the only general subclass of equations in free groups for which such results are known. We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self-contained; we assume only knowedge of basic facts about free groups.