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Robert A. Wilson
August 13, 2010

### Abstract

We give a description of the double cover of Conway's group in terms of right multiplications by 3 × 3 matrices over the octonions. This leads to simple sets of generators for many of the maximal subgroups, including a uniform construction of the Suzuki chain of subgroups.

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Johanna Rämö
July 21, 2010

### Abstract

Suppose that both n and q are even. We show that the finite simple group is strongly real if and only if 4| n . We also prove that the unipotent elements in the finite simple group are strongly real.

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Martin W. Liebeck, Alastair J. Litterick, Claude Marion
May 30, 2010

### Abstract

We produce a rigid triple of classes in the algebraic group G 2 in characteristic 5, and use it to show that the finite groups G 2 (5 n ) are not (2, 5, 5)-generated.

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Uzy Hadad
August 13, 2010

### Abstract

We prove that the length of the shortest identity in a finite simple group of Lie type of rank r defined over 𝔽 q is bounded (from above and below) by explicit polynomials in q and r .

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Tim Dokchitser
August 13, 2010

### Abstract

Given a finite group G , we address the following question: which multiples of the trivial representation are linear combinations of inductions of trivial representations from proper subgroups of G ? By Solomon's induction theorem, all multiples have this property if G is not quasi-elementary. We complement this by showing that all multiples of p are if G is p -quasi-elementary and not cyclic, and that this is best possible.

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Kay Magaard, Hung P. Tong-Viet
August 13, 2010

### Abstract

Let N be a minimal normal nonabelian subgroup of a finite group G . We will show that there exists a nontrivial irreducible character of N of degree at least 5 which is extendible to G . This result will be used to settle two open questions raised by Berkovich and Mann, and Berkovich and Zhmud'.

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Michael J. J. Barry
September 9, 2010

### Abstract

We describe a new algorithm for decomposing tensor products of indecomposable KG -modules into a direct sum of indecomposable KG -modules when K is a field of finite characteristic p and G a cyclic group of order q = p t . We use this algorithm to extend reciprocity results of Gow and Laffey relating the exterior and symmetric squares of indecomposable modules when p is odd.

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Abdullah M. Al-Roqi
July 21, 2010

### Abstract

Let G be a soluble group of odd order generated by the conjugacy class of an element g of prime order p . Let V be a faithful G -module over any field and C V ( g ) be the fixed-point subspace of g on V . We prove that .

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Danila O. Revin, Evgeny P. Vdovin
July 21, 2010

### Abstract

We obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series. As a corollary we provide a solution to Problem 5.65 from the Kourovka notebook.

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Martino Garonzi, Attila Maróti
July 21, 2010

### Abstract

For a non-cyclic finite group X let σ ( X ) be the least number of proper subgroups of X whose union is X . Precise formulas or estimates are given for σ ( S ≀ C m ) for certain non-abelian finite simple groups S where C m is a cyclic group of order m .

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A. O. Asar
August 13, 2010

### Abstract

Totally imprimitive p -groups satisfying the cyclic-block property are investigated. It is shown that in these groups any two blocks either are disjoint or one is contained in the other, and so the set of all blocks of the same size forms just one block system. Furthermore the non- FC -subgroups of these groups are transitive. For each prime p totally imprimitive p -subgroups of FSym(ℕ*) satisfying the cyclic-block property are constructed, which are not minimal non- FC -groups.

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Valeriy G. Bardakov, Paolo Bellingeri
May 30, 2010

### Abstract

We define and study extensions of the Artin and Perron–Vannier representations of braid groups to topological and algebraic generalizations of braid groups. We provide faithful representations of braid groups of oriented surfaces with boundary as automorphisms of finitely generated free groups. The induced representations of such groups as outer automorphisms of finitely generated free groups are still faithful. Also we give a representation of braid groups of closed surfaces as outer automorphisms of finitely generated free groups. Finally, we provide faithful representations of Artin–Tits groups of type 𝒟 as automorphisms of free groups.

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Yves Cornulier, Aditi Kar
August 13, 2010

### Abstract

We characterize permutational wreath products with Property (FA). For instance, we show that the standard wreath product A ≀ B of two non-trivial countable groups A and B has Property (FA) if and only if B has Property (FA) and A is a finitely generated group with finite abelianization. We also prove an analogous result for hereditary Property (FA). On the other hand, we prove that many wreath products with hereditary Property (FA) are not quotients of finitely presented groups with the same property.