Related papers: Inverse semigroups with idempotent-fixing automorp…

Let $G$ be a finite group and $\sigma=\{\sigma_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}=\emptyset$ for all $i\neq j$. A chief factor…

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is…

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing…

An automorphism of a group is said to be normal if it preserves each normal subgroup. In this paper, we determine the normal automorphisms of a free metabelian nilpotent group.

We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set then there is a common fixed point for all elements of $F.$ We also show that if $F$ is…

Let G be a profinite group. The following results are proved. The commutator subgroup G' is finite if and only if G is covered by countably many abelian subgroups. The group G is finite-by-nilpotent if and only if G is covered by countably…

The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this…

We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free…

We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\mathcal J$ of intervals of some totally ordered abelian group, these…

Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new…

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of…

For a group G and an element a in G let |a|_k denote the cardinality of the set of commutators [a,x_1,...,x_k], where x_1,...,x_k range over G. The main result of the paper states that a group G is finite-by-nilpotent if and only if there…

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly…

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and…

An automorphism $\alpha$ of a group $G$ is said to be central if $\alpha$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter…

We consider word automaticity for groups that are nilpotent of class $2$ and have exponent a prime $p$. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special…

For a finitely generated free group F_n, of rank at least 2, any finite subgroup of Out(F_n) can be realized as a group of automorphisms of a graph with fundamental group F_n. This result, known as Out(F_n) realization, was proved by…

In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of…

Around twenty years ago Ghys conjectured that finite subgroups of the diffeomorphism group of a compact smooth manifold M have an abelian normal subgroup of index at most a(M), where a(M) depends only on M. First we construct a family of…