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Morton E. Harris
February 8, 2010

### Abstract

Let N be a normal subgroup of a finite group G such that b is a block of defect zero on N . (This will be the case for any block of N if N is a p ′-group.) Let M be an indecomposable G -module with vertex Q that lies in a block of G that covers b . We present a procedure for obtaining a Green correspondent of M relative to , , and for obtaining Q -sources of M using Clifford-theoretic analyses.

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Masahiko Miyamoto
November 20, 2009

### Abstract

For a p -block B of a finite group G , it is well known that Σ χ i ∈ B χ i (1) χ i vanishes on all p -singular elements of G . The converse proposition was proposed by K. Harada and partially proved by several authors using decomposition matrices. We give a partial answer without using decomposition matrices in the case where G has a strongly p -embedded subgroup.

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Harald Ellers, John Murray
February 8, 2010

### Abstract

Let Σ n be the symmetric group of degree n , and let F be a field of characteristic p . Suppose that λ is a partition of n + 1, that α and β are partitions of n that can be obtained by removing a node of the same residue from λ , and that α dominates β . Let S α and S β be the Specht modules, defined over F , corresponding to α , respectively β . We use Jucys–Murphy elements to give a very simple description of a non-zero homomorphism S α → S β . Following Lyle, we also give an explicit expression for the homomorphism in terms of semi-standard homomorphisms. Our methods furnish a lower bound for the Jantzen submodule of S β that contains the image of the homomorphism. Our results allow us to describe completely the structure of the ring End F Σ n ( S λ ↓ Σ n ) when p ≠ 2.

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Carlo Casolo
March 10, 2010

### Abstract

Let G be a finite group acting faithfully on a finite vector space M in such a way that the centralizer of every element of M contains a Sylow q -subgroup of G as a central subgroup (for a fixed prime divisor q of | G | with ( q , | M |) = 1). Then G is isomorphic to a subgroup of the semi-linear group on M .

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David Chillag, Silvio Dolfi
February 8, 2010

### Abstract

An element x of a finite group G is called rational if all generators of the group 〈 x 〉 are contained in a single conjugacy class. If all elements of G are rational, then G itself is called rational. It was proved by Gow that if G is a rational solvable group then π ( G ) ⊂ {2, 3, 5}. We call x ∈ G semi-rational if all generators of 〈 x 〉 are contained in a union of two conjugacy classes. Furthermore, we call x ∈ G inverse semi-rational if every generator of 〈 x 〉 is conjugate to either x or x –1 . Then G is called semi-rational (resp. inverse semi-rational) if all elements of G are semi-rational (resp. inverse semi-rational). We show that if G is semi-rational and solvable then π ( G ) ⊂ {2, 3, 5, 7, 13, 17}, and if G is inverse semi-rational and solvable then 17 ∉ π ( G ). If G has odd order, then it is semi-rational if and only if it is inverse semi-rational. In this case we describe the structure of G .

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Zvonimir Janko
February 8, 2010

### Abstract

We determine here the groups of the title up to isomorphism. As a result we obtain three infinite classes of 2-groups and an exceptional group of order 2 5 .

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Songliang Chen, Yun Fan
February 8, 2010

### Abstract

We study conditions for the supersolvability of a finite group under the assumption that all subgroups of order p (or all subgroups of order 4 if p = 2) in a Sylow p -subgroup P are contained in the hyperquasicenter of the normalizer of P .

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Jing Xu
February 8, 2010

### Abstract

In this paper, we begin a classification of the vertex-transitive tournaments of order pq where p and q are distinct odd primes. In particular we characterize the pq -circulant tournaments; see Theorem 4.3. Moreover, we determine 2-closed (in Wielandt's sense) odd-order transitive permutation groups of degree p and pq by using the classifications of vertex-transitive tournaments of order p and pq . We prove that each such 2-closed group containing a cyclic regular subgroup is the full automorphism group of some circulant tournament.

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Francesco Catino, Ernesto Spinelli
February 8, 2010

### Abstract

Let KG be a non-commutative group algebra of a torsion nilpotent group G over a field K of positive characteristic p whose unit group, 𝒰( KG ), is solvable. In the present note we prove that dl(𝒰( KG )) ⩾ ⌈log 2 ( p + 1)⌉ and characterize group algebras for which this lower bound is achieved.

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Andrea Lucchini
February 8, 2010

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A. Clifford, R. Z. Goldstein
February 8, 2010

### Abstract

An algorithm is developed to determine whether a subgroup of a free group contains a primitive element This answers question 39b in the list of problems on free groups posted on the World of Groups website, www.grouptheory.info.

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Luis Paris
February 8, 2010

### Abstract

We prove that the mapping class group of a closed oriented surface of genus ρ ⩾ 3 has no proper subgroup of index ⩽ 4 ρ + 4.