Related papers: A focal subgroup theorem for outer commutator word…

We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite…

For a Baumslag-Solitar group $G$ we calculate the intersection $\gamma_w(G)$ of all terms of the lower central sequence of $G$.Using this we are able to show that $[\gamma_w(G),G]=\gamma_w(G)$ thus answering a question of Bardakov and…

For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct…

A pro-p Cappitt group is a pro-p group G such that the subgroup topologically generated by all non-normal closed subgroups is a proper subgroup of G. In this paper we prove that non-abelian pro-p Cappitt groups whose torsion subgroup is…

A subgroup $H$ of a finite group $G$ is called submodular in $G$, if we can connect $H$ with $G$ by a chain of subgroups, each of which is modular (in the sense of Kurosh) in the next. If a group $G$ is supersoluble and every Sylow subgroup…

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpi\'nski gasket that…

We define a local Sylow subgroup of a totally disconnected, locally compact group G to be a maximal pro-p subgroup of an open compact subgroup of G. We use these subgroups to define the p-localisation of G, a locally virtually pro-p group…

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental…

Let $p$ be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd $p$-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture and the Alperin-McKay…

We show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general…

A Sylow p-subgroup P of a finite group G is called redundant if every p-element of G lies in a Sylow subgroup different from P. Generalizing a recent theorem of Mar\'oti--Mart\'inez--Moret\'o, we show that for every non-cyclic p-group P…

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that…

Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $\langle H,H^g\rangle\cap N_G(H)=H$ for all $g\in G.$ In this paper, we determine the structure of finite groups with an extremely…

For a given m>=1, we consider the finite non-abelian groups G for which |C_G(g):<g>|<=m for every g in G\Z(G). We show that the order of G can be bounded in terms of m and the largest prime divisor of the order of G. Our approach relies on…

Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and…

We prove that the conjugacy problem in Out(Fm) is solvable for the class of outer automorphisms whose restrictions to their polynomial subgroups are of finite order. To do this, we first investigate the structure of suspensions of free…

In this article, we shall generalize a theorem due to Frobenius in group theory, which asserts that if $p$ is a prime and $p^{r}$ divides the order of a finite group, then the number of subgroups of order $p^{r}$ is $\equiv$ 1(mod $p$).…

Waring's classical problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for nonabelian groups, and simple groups in particular. Here the k-th…

The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We…

We show that, given a continuous action $\alpha$ of a locally compact group $G$ on a factor $M$, the relative commutant $M'\cap(M\rtimes_{\alpha} G)$ is contained in $M\rtimes_{\alpha} H$ where $H$ is the subgroup of elements acting without…