*Abstract*: Let be the average order of the elements of , where is a finite group. We show that there is no polynomial lower bound for in terms of , where , even when is a prime-power order group and is abelian. This gives a negative answer to a question of A. Jaikin-Zapirain.

Abstract: Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that N_w(1) \geq |G|^{k-1}, where N_w(1) is the number of k-tuples (g_1,…,g_k) of elements in G such that w(g_1,…,g_k)=1. This conjecture is known to be true for finite groups of nilpotency class 2. In this talk, we consider a generalized version of Amit’s conjecture and discuss known results.

Title: In search of new hereditarily just infinite groups

Abstract: An infinite group is said to be just infinite if it has no infinite proper quotient. A just infinite group is hereditarily just infinite if each of its finite index subgroups is just infinite.

Hereditarily just infinite groups are – as of today – quite mysterious. In this talk we will survey what is know about them and I will present some work in progress with G. Fernandez-Alcober and M. Noce about hereditarily just infinite groups representable as self-similar subgroups of rooted trees.

The first seminar will be held on Thursday, October 29, at 16:00 Novosibirsk time (=9:00 London time) via ZOOM. The first talk is by **Evgeny Khukhro**, *Some problems about bounding length parameters of finite groups*. The zoom link is https://us02web.zoom.us/j/5578968779

This series of seminars is intended to highlight the most promising research directions in group theory and, more generally, algebra, along with other relevant fields of mathematics. Clear formulations of the problems may help young researchers to focus their efforts and give them a higher chance of achieving outstanding mathematical results.

The seminars will be held once in two weeks on Thursdays. The starting time of the seminar will vary depending on the preferences of the speaker, taking into account the geography of the audience.

]]>*Abstract*: A right Engel sink of an element of a group is a set such that for every all sufficiently long commutators belong to . (Thus, is a right Engel element precisely when we can choose .) It is proved that if every element of a compact (Hausdorff) group has a countable right Engel sink, then has a finite normal subgroup such that is locally nilpotent.