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Helge Glöckner
August 17, 2006

### Abstract

We analyze the structure of locally compact groups which can be built up from p -adic Lie groups, for p in a given set of primes. In particular, we calculate the scale function and determine tidy subgroups for such groups, and use them to recover the primes needed to build up the group.

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Dessislava H Kochloukova, Aline G. S Pinto
August 17, 2006

### Abstract

Let G be a (topologically) finitely generated metabelian pro- p group. We prove the conjecture of J. King [7] that for any natural number m the group G embeds in a metabelian pro- p group of type FP m over ℤ p .

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Jeffrey Burdges
August 17, 2006

### Abstract

The algebraicity conjecture for simple groups of finite Morley rank, also known as the Cherlin–Zil'ber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last fifteen years, the main line of attack on this problem has been Borovik's program of transferring methods from finite group theory, which has led to considerable progress; however, the conjecture itself remains completely open. In Borovik's program, groups of finite Morley rank are divided into four types, odd, even, mixed, and degenerate, according to the structure of their Sylow 2-subgroup. For even and mixed type the algebraicity conjecture has been proven.

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Ivan Marin
August 17, 2006

### Abstract

We show that the residual nilpotence of pure Artin groups of spherical type is easily deduced from the faithfulness of the Krammer–Digne representations.

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Carl Droms
August 17, 2006

### Abstract

We find necessary and sufficient conditions for a finitely generated group with more than one end to have a planar Cayley graph.

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Jon M Corson, Thomas J Ratkovich
August 17, 2006

### Abstract

In general the extension of a residually finite group by a residually finite group may not be residually finite. We define a strong form of residual finiteness for groups and show that the property is closed under extensions. We then show how groups with this property can be constructed using amalgamated free products and HNN extensions. This leads to a multitude of examples.

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L. A Kurdachenko, A Russo, G Vincenzi
August 17, 2006

### Abstract

A subgroup H of a group G is said to be abnormal in G if g ∈ < H , H g > for each element g ∈ G . It is well known that every locally nilpotent group has no proper abnormal subgroups, but it is an open question whether the converse holds. In this article we prove this conjecture for some classes of infinite groups. In particular, it is proved that an FC -nilpotent group without proper abnormal subgroups is hypercentral. Also groups with finitely many abnormal subgroups are considered.

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Primož Moravec
August 17, 2006

### Abstract

A group G is said to be n -central if the factor group G/Z ( G ) is of exponent n . We improve a result of Gupta and Rhemtulla by showing that every 4-central group is 16-abelian and every 6-central group is 36-abelian. There are examples of finite groups which show that these bounds are best possible. Consequently, we can completely describe the structure of exponent semigroups of free non-cyclic n -central groups for n = 2, 3, 4, 6. We obtain a characterization of metabelian p -central groups and a classification of finitely generated 2-central groups. We compute the nilpotency class of the free metabelian 4-central group of arbitrary finite rank.

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John N Bray, Robert A Wilson
August 17, 2006

### Abstract

In the Kourovka Notebook, Deaconescu asks if |Aut G | ≥ φ(| G |) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G | = φ(| G |). In an earlier paper we have answered both questions in the negative, and shown that |Aut G |/φ(| G |) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. The problem remains open when G is supersoluble, or nilpotent.

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John R Britnell
August 17, 2006

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