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September 13, 2006
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The articles [R. Gramlich, C. Homan and S. Shpectorov. A Phan-type theorem for Sp (2 n; q). J. Algebra 264 (2003), 358–384.] and [R. Gramlich. Weak Phan systems of type C n . J. Algebra 280 (2004), 1–19.] give a characterization of central quotients of the group Sp(2 n , q ) for n ≥ 3 and all prime powers q , up to some small cases that are left open. The present article fills in this gap, thus providing the definitive version of the Phan-type theorem for Sp(2 n , q ).

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It is shown that, with only a few small exceptions, any finite symmetric group is a faithful completion of the Goldschmidt amalgams of type G 1 1 , G 1 2 , and G 1 3 . A recursive procedure is used to ‘add’ copies of small completions, in particular that of Sym(9), to obtain nine infinite series of completions isomorphic to symmetric groups.

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Let Γ be a finite locally ( G , s )-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G -orbits. In previous work the authors showed that if G has a non-trivial normal subgroup intransitive on both of the vertex orbits of G , then Γ is a cover of a smaller locally s -arc transitive graph. Thus the ‘basic’ graphs to study are those for which G acts quasiprimitively on at least one of the two orbits. In this paper we investigate the case where G is quasiprimitive on only one of the two G -orbits. Such graphs have a normal quotient which is a star. We construct several infinite families of locally 3-arc transitive graphs and prove characterization results for several of the possible quasiprimitive types for G .

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The main result of this paper is a decomposition theorem for the exterior square of an indecomposable module of a cyclic 2-group in characteristic 2. We also obtain a reciprocity theorem relating the exterior and symmetric squares for indecomposable modules of a cyclic p -group in odd characteristic.

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Let R be a commutative ring with unity and let G be a group. The group ring RG has a natural involution that maps each element of G to its inverse. We denote by RG + the set of symmetric elements under this involution. We study necessary and suffient conditions for RG + to be commutative or, equivalently, for RG + to be a subring of RG . We also determine all torsion groups G such that the set of symmetric units of RG is a subgroup, when char( R ) is an odd prime number.

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We show that for any finite group G and for any d there exists a word w ∈ F d such that a d -tuple in G satisfies w if and only if it generates a solvable subgroup. As a corollary, the probability that a word is satisfied in a fixed non-solvable group can be made arbitrarily small; this answers a question of Alon Amit. It also follows that there is no absolute bound in the Baumslag–Pride theorem for the minimal index in a group with at least two more generators than relators of a subgroup that can be mapped homomorphically onto a non-abelian free group.

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For a fixed word hyperbolic group we compare different residual properties related to quasiconvex subgroups.