Let G be a finite group and C2 the cyclic group of order 2. Consider the 8 multiplicative operations (x, y) 7→ (xiyj)k, where i, j, k ∈ {−1, 1}. Define a new multiplication on G × C2 by assigning one of the above 8 multiplications to each quarter (G × {i}) × (G × {j}), for i, j ∈ C2. When G is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomo...

Conjecture 1 (A. Potapchik and A. Rapinchuk). Let D be a finite dimensional division algebra over an arbitrary field. Then D# does not have any normal subgroup N such that D#/N is a nonabelian finite simple group. Of course D# is the multiplicative group of D. Conjecture 1 appears in [4]. It is related to the following conjecture of G. Margulis and V. Platonov (Conjectures 9.1 and 9.2, pages 51...

This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is...

This paper introduces Markov chains and processes over nonabelian free groups and semigroups. We prove a formula for the f -invariant of a Markov chain over a free group in terms of transition matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the AbramovRohlin formula for skew-product actions and Yuzvinskii’s addition form...

In previous work we showed that the only finite nonabelian simple group acting by diffeomorphisms on a homology 3sphere is the alternating or dodecahedral group A5. Here we characterize finite nonsolvable groups that act on a homology 3-sphere preserving orientation. We find exactly the finite nonsolvable groups that act orthogonally on the 3-sphere, plus two families of groups for which we do ...

Given a finite group G, let PG(s) be the probability that s randomly chosen elements generate G, and let H be a finite group with PG(s) = PH(s). We show that if the nonabelian composition factors of G and H are PSL(2, p) for some non-Mersenne prime p ≥ 5, then G and H have the same non-Frattini chief factors. Mathematics Subject Classification (2010). 20D06.

It is suggested that a unifying description of leptons and hadrons can be obtained within a nonabelian gauge theory where the gauge group is a symmetry group of a set of massless elementary fermions (leptons, quarks). We investigate the consequences of such an approach for the strong, electromagnetic, and weak interactions. We study both gauge theories with and without fermion number conservati...

Given a finite non-cyclic group G, call σ(G) the smallest number of proper subgroups of G needed to cover G. Lucchini and Detomi conjectured that if a nonabelian group G is such that σ(G) < σ(G/N) for every non-trivial normal subgroup N of G then G is monolithic, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct stud...

In a group trellis, the sequence of branches that split from the identity path and merge to the identity path form two normal chains. The Schreier refinement theorem can be applied to these two normal chains to obtain a Schreier series, a normal chain of the branch group B of the group trellis. It is shown that the components of the shortest length generator sequences of Forney and Trott form a...