Volume 15, Issue 6

Abstract. We study the 14-dimensional real representation of the finite simple group G 2 (3) and relate it to (a) the compact real form of the Lie group G 2 , (b) the smallest Ree group and (c) representations in characteristic 3. In particular, we give a set of generators which leads to a new and easy proof that the group is indeed G 2 (3).

Abstract. For an element g in a group X , we say that g has 2-part order if is the largest power of 2 dividing the order of g . Using results of Erdős and Turán, and Beals et al., we give explicit lower bounds on the proportion of elements of the symmetric group with certain 2-part orders. Some of these lower bounds are constant; for example we show that at least 23.5% of the elements in () have a certain 2-part order and furthermore, more than half of the elements in have one of three 2-part orders. Also, for all , at least of the elements in have the same 2-part order and we show that is best possible.

Abstract. We study the reduced Lefschetz module of the complex of p -centric and p -radical subgroups. We assume that the underlying group G has parabolic characteristic p and the centralizer of a certain noncentral p -element has a component with central quotient H a finite group of Lie type in characteristic p . A nonprojective indecomposable summand of the associated Lefschetz module lies in a nonprincipal block of and it is a Green correspondent of an inflated, extended Steinberg module for a Lie subgroup of H . The vertex of this summand is the defect group of the block in which it lies. The application of these results to sporadic finite simple groups yields nine groups when and eight groups when for which the reduced Lefschetz module has precisely one nonprojective summand.

Abstract. A problem proposed by Grishkov asks if every metabelian triality group has a corresponding Moufang loop which is a group. Here we show that if minimal counter-examples exist, then such triality groups have to be p -groups. In fact, for any prime p , there exists a metabelian p -group that admits triality and has a corresponding Moufang loop that is not a group. Here we also determine how to strengthen the hypothesis of the original problem so that the corresponding Moufang loop is indeed a group.

Abstract. A subgroup of a polycyclic-by-finite group is hypercentrally embedded if and only if its projections into the finite quotients have this property.

Abstract. A group G is said to be m -rigid if it has a normal series in which each factor is abelian and torsion-free as a -module. Denote by the class of all m -rigid groups and by the set of groups in generated by that satisfy a given set of relations R . We say that a group in is maximal if it has no proper covering in . It is proved that, for every R , the set contains only finitely many maximal groups. The set of relations R is said to be complete if contains a unique maximal group. It is shown that every finitely generated group in is completely finitely presented. We give a definition of a canonical presentation for a rigid group with the generators . If such a presentation is given, the group at least has decidable word problem. Given a finite set of relations , we effectively construct a finite set of canonical presentations in the generators for groups in among which all the maximal groups in are contained.

Abstract. Let be a right-angled Artin group. We use geometric methods to compute a presentation of the subgroup of consisting of the automorphisms that send each generator to a conjugate of itself. This generalizes a result of McCool on basis-conjugating automorphisms of free groups.

Abstract. We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group.

Abstract. Let T be a tree and e an edge in T . If C is a component of and both C and its complement are infinite, we say that C is a half-tree. The main result of this paper is that if G is a closed subgroup of the automorphism group of T and G leaves no non-trivial subtree invariant and fixes no end of T , then the subgroup generated by the pointwise stabilizers of half-trees is topologically simple. This result is used to derive analogues of recent results of Caprace and De Medts (2011) and it is also applied in the study of the full automorphism group of a locally finite primitive graph with infinitely many ends.