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In 1999 Orin Chein and Andrew Rajah [Comment. Math. Univ. Carolin. 41: 237–244, 2000] presented the following question. If a Moufang loop G contains a normal abelian subgroup N of odd order such that G / N is cyclic, must G be a group? Here we prove that a Moufang loop that is an extension of an abelian group of odd order by a cyclic group has a normal subgroup of index dividing 3.

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence 𝒦 = {( G i , 1) : i ∈ ℕ} consisting of finite simple subgroups. Then for any finite subgroup F consisting of 𝒦-semisimple elements in G , the centralizer C G ( F ) has an infinite abelian subgroup A isomorphic to a direct product of ℤ p i for infinitely many distinct primes p i . Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence 𝒦 = {( G i , 1) : i ∈ ℕ} consisting of finite simple subgroups G i and F is a finite 𝒦-semisimple subgroup of G , then C G ( F ) involves an infinite simple non-linear locally finite group provided that the finite fields k i over which the simple group G i is defined are splitting fields for L i , the inverse image of F in Ĝ i for all i ∈ ℕ. The group Ĝ i is the inverse image of G i in the corresponding universal central extension group.

In this paper we study G -arc-transitive graphs Δ where the permutation group induced by the stabiliser G x of the vertex x on the neighbourhood Δ( x ) satisfies the two conditions given in the introduction. We show that for such a G -arc-transitive graph Δ, if ( x, y ) is an arc of Δ, then the subgroup of G fixing Δ( x ) and Δ( y ) point-wise is a p -group for some prime p . Next we prove that every G -locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-like theorem for a very large class of arc-transitive graphs. Furthermore, we give various families of G -arc-transitive graphs where our two local conditions do not apply and where has arbitrarily large composition factors.

We prove that Thompson's group F is not minimally almost convex with respect to any generating set which is a subset of the standard infinite generating set for F and which contains x 1 . We use this to show that F is not almost convex with respect to any generating set which is a subset of the standard infinite generating set, generalizing results in [Horak, Stein, and Taback, Int. J. Algebra Comput. 19: 963–997, 2009].

Let K be a global field and S be a finite set of places of K which includes all those of archimedean type. Let G be an algebraic group over K and G K be its K -rational points. The authors provide a detailed proof of a lemma of Raghunathan which states that (under fairly weak restrictions) the closure in the S -congruence topology of a subgroup of G K normalized by an S -arithmetic subgroup is also open. This leads to a significant simplification in the proof of one of the principal results in a recent joint paper of the authors. By applying the lemma to S -arithmetic lattices in G of K -rank one, where char( K ) ≠ 0 and | S | = 1, we can provide a lower estimate for the number of subgroups of a given index in such a lattice which are not S -congruence. This extends previous results of the first author and Andreas Schweizer.

For n ⩾ 4, we describe Artin's braid group on n strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations.

Let G be a finite group. Goldschmidt, Flores and Foote investigated the following concept: A subgroup A of S is said to be strongly closed in S with respect to G if for every a ∈ A , every element of S that is fused in G to a lies in a G . In particular, when A is of prime-power order and S is a Sylow subgroup containing it, A is simply said to be a strongly closed subgroup. Bianchi, Mauri, Herzog and Verardi called a subgroup H of G an ℋ-subgroup if N G ( H ) ∩ H g ⩽ H for all g ∈ G . In fact, an ℋ-subgroup of prime power order is the same as a strongly closed subgroup. In this paper, groups with certain ℋ-subgroups of prime order are studied. For example, we give new characterizations of finite minimal non--groups (a -group is a group for which normality is a transitive relation) by ℋ-subgroups.

Motivated by the well-known conjecture of Andrews and Curtis [Amer. Math. Monthly 73: 21–28, 1966.], we consider the question of how, in a given n -generator group G , any ordered n -tuple of “annihilators” of G , that is, with normal closure all of G , can be transformed by standard moves into a generating n -tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (“elementary M-transformations”) by means of which every annihilating n -tuple can be transformed into a generating n -tuple. We obtain upper estimates for the recalcitrance of n -generator finite groups—thus quantifying a result from [Borovik, Lubotzky, and Myasnikov, Progr. Math. 248: 15–30, 2005]—and of a wide class of n -generator solvable groups, thus extending and correcting a result from [Burns, Herfort, Kam, Macedońska, and Zalesskii, Bull. Austral. Math. Soc. 60: 245–251, 1999].

In this paper, we generalize portions of the theory of localization to the category of nilpotent R -powered groups, where R is a binomial UFD. In particular, we show that if ω is a set of primes in R and G is a finitely R -generated nilpotent R -powered group, there exists a unique nilpotent R -powered ω -local group that is, in some sense, the “best approximation” to G among all nilpotent R -powered ω -local groups. We also use various residual properties of nilpotent R -powered groups to prove that every ω -localization map is an ω -isomorphism when R is a PID containing ℚ and G is finitely R -generated.

It is known that Moufang loops are closely related to groups as they have many properties in common. For instance, they both have an inverse for each element and satisfy Lagrange's theorem. In this paper, we study the properties of a class of Moufang loops which are not groups, but all their proper subloops and proper quotient loops are groups.