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Chris Parker, Gernot Stroth
May 2, 2013

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Abstract. We identify the sporadic groups M(23) and F 2 from the approximate structure of the centralizer of an element of order 3.

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Gareth A. Jones
May 2, 2013

### Abstract

Abstract. A Beauville surface of unmixed type is a complex algebraic surface which is the quotient of the product of two curves of genus at least 2 by a finite group G acting freely on the product, where G preserves the two curves and their quotients by G are isomorphic to the projective line, ramified over three points. We show that the automorphism group A of such a surface has an abelian normal subgroup I isomorphic to the centre of G , induced by pairs of elements of G acting compatibly on the curves (a result obtained independently by Fuertes and González-Diez). Results of Singerman on inclusions between triangle groups imply that A / I is isomorphic to a subgroup of the wreath product , so A is a finite solvable group. Using constructions based on Lucchini's work on generators of special linear groups, we show that every finite abelian group can arise as I , even if one restricts the index to the extreme values 1 or 72.

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Stefanos Aivazidis
May 2, 2013

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Abstract. The subgroup permutability degree of a group G is defined as the probability that two subgroups of G permute, or equivalently that the product of two subgroups is again a subgroup. Tărnăuceanu asked in “Addendum to `Subgroup commutativity degrees of finite groups'” (J. Algebra 337 (2011), 363–368) whether there exist families of groups other than dihedral, quasi-dihedral or generalised quaternion (all of 2-power cardinality), whose subgroup permutability degree tends to 0 as the size of the group tends to infinity. The purpose of this paper is to provide an affirmative answer for the family of projective special linear groups over fields of even characteristic by utilising a result of Dickson that gives a complete list of the subgroups of PSL 2 ( q ).

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Conchita Martínez-Pérez, Wolfgang Willems
May 2, 2013

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Abstract. We discuss the structure of finite groups for which the projective indecomposable modules have special given dimensions. In particular, we prove the converse of Fong's dimension formula for p -solvable groups. Furthermore, we characterize groups for which all irreducible p -Brauer characters have p -power degrees.

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Alonso Castillo-Ramirez
May 2, 2013

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Abstract. The concept of Majorana representation was introduced by A. A. Ivanov (2009), as a tool to identify and study subalgebras of the Conway–Griess–Norton Monster algebra . Sakuma's theorem in “6-transposition property of -involutions of vertex operator algebras”, Int. Math. Res. Not. IMRN 2007 (2007), Article ID rnm030, states that there are eight possibilities for the isomorphism type of an algebra with scalar product generated by a pair of distinct Majorana axes. These algebras, now known as the Norton–Sakuma algebras, were described by S. P. Norton (1982) as 2-generated subalgebras of and labelled by types , , , , , , and . In the present paper, we contribute to the understanding of the Norton–Sakuma algebras by finding all their idempotent elements and their automorphism groups. In particular, we find that an algebra of type has infinitely many idempotents of length 2, and that an algebra of type has exactly 208 idempotents.

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Anthony M. Gaglione, Seymour Lipschutz, Dennis Spellman
May 2, 2013

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Abstract. Let G be a free metabelian group. Let C be the centralizer in G of an element . Let be the corresponding free rank 1 centralizer extension of G relative to the variety of metabelian groups. We find a matrix representation and prove that is discriminated by epimorphisms . We conjecture that ρ is faithful in the case .