Further variations on the theme of FC-groups

Groups that are FC, or more generally satisfy any of the weakenings of the FC-condition considered in de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et?al. (J. Algebra 326:218?C226, 2011), have local systems consisting of normal finite-by-nilpotent subgroups. Apart from generalizing results from de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et al. (J. Algebra 326:218?C226, 2011) to the more general context of locally (normal and finite-by-nilpotent) groups, we partially settle an open problem raised in Robinson et?al. (J. Algebra 326:218?C226, 2011) concerning the isomorphism of maximal p-subgroups, but in this more general setting of locally (normal and finite-by-nilpotent) groups.     

On just infinite periodic locally soluble groups

We construct an uncountable family of periodic locally soluble groups which are hereditarily just infinite. We also show that the associated full \(C^*\)-algebra \(C^*(G)\) is just infinite for many groups G in this family.     

On the proportion of locally soluble superelliptic curves

We investigate the proportion of superelliptic curves that have a ${\mathbb{Q}}_p$ point for every place p of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3=f(x,z)$ for an integral binary form f of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in p for the proportion of such curves over ${\mathbb{Z}}_p$ having a ${\mathbb{Q}}_p$-point.     

Free actions of virtually FC-groups

If an infinite group G admits a free action by a group of automorphisms A which is virtually an FC-group and which has only finitely many orbits, then G is isomorphic to the additive group of a field and the action is that of a group of semilinear transformations. Received: 21 February 2005     

Variations on the theme of FC-groups

The following is clearly equivalent to the usual definition of FC-group. A group is an FC-group, if each of its cyclic subgroups has only finitely many conjugates. We consider several weaker conditions on the conjugates of cyclic subgroups, the strongest of which we show is equivalent to the FC-condition for many classes of groups.     

Injectors of finite solvable groups

It is well known that in a free group the simple commutator [Y, X](alternatively called the first Engel word) can be expressed as a product of squares. Likewise the second Engel word [Y, X, X] can be expressed as a product of cubes. Results on groups of exponent four imply that the fifth Engel word [Y, X, X, X, X, X] can be expressed as a product of fourth powers, and explicit expressions have now been obtained with the assistance of a computer. The results and the computeraided technique are described.