Related papers: On (hereditarily) just infinite profinite groups t…
A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is…
We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are 'universal' in that all (hereditarily) just infinite profinite groups arise…
A profinite group $G$ is just infinite if every closed normal subgroup of $G$ is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup $H$ of $G$, there are only finitely many…
We study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many…
For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p…
Examples are given of profinite groups that are not strongly complete, and have other `bad' properties, yet have only finitely many open subgroups of each finite index. It is shown that a profinite group with the latter property must be…
We show that the group constructed in the 2004 paper 'A 2-generated just-infinite profinite group which is not positively finitely generated' by A. Lucchini is in fact hereditarily just infinite and contains every countably based profinite…
Just infinite groups play a significant role in profinite group theory. For each $c \geq 0$, we consider more generally JNN$_c$F profinite (or, in places, discrete) groups that are Fitting-free; these are the groups $G$ such that every…
We obtain some general restrictions on the continuous endomorphisms of a profinite group G under the assumption that G has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if G is…
A group $G$ is said to have restricted centralizers if for every $x\in G$ the centralizer $C_G(x)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we…
For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact…
The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every…
The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…
A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a…
A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of…
We investigate some properties of the $p$-elements of a profinite group $G$. We prove that if $p$ is odd and the probability that a randomly chosen element of $G$ is a $p$-element is positive, then $G$ contains an open prosolvable subgroup.…
We define the notion of accessibility for a pro-$p$ group. We prove that finitely generated pro-$p$ groups are accessible given a bound on the size of their finite subgroups. We then construct a finitely generated inaccessible pro-$p$…
Continuing recent studies of both the hereditary and super properties of certain classes of Abelian groups, we explore in-depth what is the situation in the quite large class consisting of directly finite Abelian groups. Trying to connect…
A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take…
Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups.