GROUP ALGEBRAS WHOSE UNIT GROUP IS LOCALLY NILPOTENT

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units $V(FG)$ of the group algebra $FG$ is locally nilpotent; (ii) the set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

On (𝑛 + ½)-Engel groups

Let n be a positive integer. We say that a group G is an {(n+\frac{1}{2})}-Engel group if it satisfies the law {[x,{}_{n}y,x]=1}. The variety of {(n+\frac{1}{2})}-Engel groups lies between the varieties of n-Engel groups and {(n+1)}-Engel groups. In this paper, we study these groups, and in particular, we prove that all {(4+\frac{1}{2})}-Engel {\{2,3\}}-groups are locally nilpotent. We also show that if G is a {(4+\frac{1}{2})}-Engel p-group, where {p\geq 5} is a prime, then {G^{p}} is locally nilpotent.

Groups which satisfy a Thue–Morse identity

In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity [Formula: see text]. We prove that if [Formula: see text] is a 3-Thue–Morse group then [Formula: see text] is soluble for every [Formula: see text] and [Formula: see text] in [Formula: see text]. Furthermore, if [Formula: see text] is an Engel group without involution then we show that [Formula: see text] is locally nilpotent.

Sub-Finsler structures on the Engel group

A one-parameter family of left-invariant sub-Finsler problems on a four-dimensional nilpotent Lie group of depth 3 with two generators is considered. The indicatrix of sub-Finsler structures is a square rotated by an arbitrary angle in the distribution. Methods of optimal control theory are applied. Abnormal and singular normal trajectories are described, and their optimality is proved. Singular trajectories arriving at the boundary of the reachable set in fixed time are characterized. A bang-bang phase flow is constructed, and estimates for the number of switchings on bang-bang trajectories are obtained. The structure of all normal extremals is described. Mixed trajectories are studied.