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Donald L. White
January 3, 2013

### Abstract

Abstract. Denote by S the projective special linear group PSL 2 ( q ) over the field of q elements. We determine, for all values of , the degrees of the irreducible complex characters of every group H such that . We also determine the character degrees of certain extensions of the special linear group SL 2 ( q ). Explicit knowledge of the character tables of SL 2 ( q ), GL 2 ( q ), PSL 2 ( q ), and PGL 2 ( q ) is used along with standard Clifford theory to obtain the degrees.

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John Ballantyne, Nicholas Greer, Peter Rowley
January 3, 2013

### Abstract

Abstract. For a group G , a set of odd positive integers and X a set of involutions of G we define a graph . This graph, called a -local fusion graph, has vertex set X with joined by an edge provided and the order of is in . In this paper we investigate when G is a finite symmetric group for various choices of X and .

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Xiaoyu Chen, Wenbin Guo
January 3, 2013

### Abstract

Abstract. For a maximal subgroup M of a finite group G , a θ̂-pair for M is any pair of subgroups ( C , D ) of G such that (i) , , (ii) , , (iii) no non-trivial proper subgroup of is normal in and (iv) either or there exists a subgroup E of G such that C is a maximal subgroup of E and is not contained in M . By using this notion, we exhibit some new characterizations on the structure of finite groups.

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Tsunenobu Asai, Naoki Chigira, Takashi Niwasaki, Yugen Takegahara
January 3, 2013

### Abstract

Abstract. For a finite group A and for a finite group G on which A acts, the number of crossed homomorphisms from A to G is a multiple of provided that B is a normal subgroup of A such that is cyclic. We prove a character-theoretic version of this fact, which was inspired by a theorem of P. Hall.

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Stefan Kohl
January 3, 2013

### Abstract

Abstract. We present a series of countable simple groups generated by involutions interchanging disjoint residue classes modulo lattices in ℤ d ().

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Jonathan D. H. Smith
January 3, 2013

### Abstract

Abstract. A classical construction associates a Bruck loop with a Moufang or Bol loop on which the squaring map is a permutation. The idea of that construction is now extended to a quadratically closed Bol loop or group (where the squaring operation is surjective) that satisfies the 2-Engel condition (for groups: elements commute with their conjugates). The Bruck loop is built on a space of sequences of elements from the original Bol loop. More generally, given a quadratically closed loop with the 2-Engel condition from the join of the varieties of diassociative and (right) Bol loops, a power-associative loop with the right inverse and automorphic inverse properties is constructed in similar fashion.

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Greg Conner, Michael Mihalik
January 3, 2013

### Abstract

Abstract. If G is a group, then subgroups A and B are commensurable if has finite index in both A and B . The commensurator of A in G , denoted by , is It is straightforward to check that is a subgroup of G . A subgroup A is commensurated in G if . The centralizer of A in G is a subgroup of the normalizer of A in G which is a subgroup of . We develop geometric versions of commensurators in finitely generated groups. In particular, iff the Hausdorff distance between A and is finite. We show a commensurated subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is commensurated iff a Schreier (left) coset graph is locally finite. The ends of this coset graph correspond to the filtered ends of the pair ( G , A ). This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary goals in this paper are to develop and compare the basic theory of commensurated subgroups to that of normal subgroups, and to initiate the development of the asymptotic theory of commensurated subgroups.

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Jonathan A. Kiehlmann
January 3, 2013

### Abstract

Abstract. In the first half of this paper, we outline the construction of a new class of abelian pro- p groups, which covers all countably-based pro- p groups. In the second half, we study these groups, and classify them up to topological isomorphism and abstract isomorphism. Topologically, any infinitely-generated abelian pro- p group decomposes as a product of infinitely many non-trivial groups. It emerges that they are all abstractly isomorphic to Cartesian products of finite groups and p -adic integers. We have thus constructed uncountably many pairwise topologically non-isomorphic profinite groups abstractly isomorphic to a Cartesian product of cyclic groups.