Search

Let 𝑝 be a prime. We produce two new families of pro-𝑝 groups which are not realizable as absolute Galois groups of fields. To prove this, we use the 1-smoothness property of absolute Galois pro-𝑝 groups. Moreover, we show in these families, one has several pro-𝑝 groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of the norm residue theorem), or the vanishing of Massey products in Galois cohomology.

Let [Formula: see text] be the simple group [Formula: see text], where [Formula: see text] is a prime number. For any subgroup [Formula: see text] of [Formula: see text], we compute the Möbius function [Formula: see text] of [Formula: see text] in the subgroup lattice of [Formula: see text]. To this aim, we describe the intersections of maximal subgroups of [Formula: see text]. We point out some connections of the Möbius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of [Formula: see text]-subgroups of [Formula: see text], for any prime [Formula: see text] and any prime power [Formula: see text].

For a prime number $p$, we give a new restriction on pro-$p$ groups $G$ which are realizable as the maximal pro-$p$ Galois group $G_F(p)$ for a field $F$ containing a root of unity of order $p$. This restriction arises from Kummer Theory and the structure of the maximal $p$-radical extension of $F$. We study it in the abstract context of pro-$p$ groups $G$ with a continuous homomorphism $\theta\colon G\to1+p\mathbb{Z}_p$, and characterize it cohomologically, and in terms of 1-cocycles on $G$. This is used to produce new examples of pro-$p$ groups which do not occur as maximal pro-$p$ Galois groups of fields as above., Comment: Final revised version. To appear in the Journal of Algebra

Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim H^1(G_{K},\mathbb{F}_p), Comment: Updated version

Let $p$ be a prime. We show that if a pro-$p$ group with at most 2 defining relations has quadratic $\mathbb{F}_p$-cohomology, then such algebra is universally Koszul. This proves the "Universal Koszulity Conjecture" formulated by J. Min\'a\v{c} et al. in the case of maximal pro-$p$ Galois groups of fields with at most 2 defining relations., Comment: To appear on "Math. Scandinavica"

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a continuous representation $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $\theta\vert_H$, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-$p$ group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-$p$ Galois groups of fields, and that if a 1-smooth pro-$p$ group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-$p$ Galois groups of fields. Finally we ask whether 1-smooth pro-$p$ groups satisfy a "Tits' alternative"., Comment: To appear on "Canadian Math. Bull"

$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $\Gamma$ are groups of order $p^{2} q$ with non-isomorphic Sylow $p$-subgroups, then there are no regular subgroups of the holomorph of $G$ which are isomorphic to $\Gamma$. Equivalently, a Galois extension with Galois group $\Gamma$ has no Hopf-Galois structures of type $G$. Our method relies on the alternate brace operation $\circ$ on $G$, which we use mainly indirectly, that is, in terms of the functions $\gamma : G \to \Aut(G)$ defined by $g \mapsto (x \mapsto (x \circ g) \cdot g^{-1})$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual., Comment: 43 pages

We introduce a generalization $G^{(\alpha)}(X)$ of the truncated logarithm $\mathcal{L}_1(X) = \sum_{k=1}^{p-1}X^k/k$ in characteristic $p$, which depends on a parameter $\alpha$. The main motivation of this study is $G^{(\alpha)}(X)$ being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials. Such Laguerre polynomials play a role in a grading switching technique for non-associative algebras, previously developed by the authors, because they satisfy a weak analogue of the functional equation $\exp(X)\exp(Y)=\exp(X+Y)$ of the exponential series. We also investigate functional equations satisfied by $G^{(\alpha)}(X)$ motivated by known functional equations for $\mathcal{L}_1(X)=-G^{(0)}(X)$., Comment: 24 pages

We consider generalisations of Thompson's group $V$, denoted by $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(\Sigma)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure., Comment: 24 pages, no figures

In this thesis we study saturated fusion systems on \(p\)-groups having sectional rank 3, for \(p\) odd. We obtain a complete classification of simple fusion systems on p-groups having sectional rank 3 for \(p\) ≥ 5, exhibiting a new simple exotic fusion system on a 7-group of order 7\(^∧\)5. We introduce the notion of pearls, defined as essential subgroups isomorphic to the groups C\(_p\) X \(_p\) and \(p\)\(_+\)\(^1\)\(^+\)\(^2\) (for odd), and we illustrate some properties of fusion systems involving pearls. As for \(p\) = 3, we determine the isomorphism type of a certain section of the 3-groups considered.

S. S. Magliveras et al. have described symmetric and public key cryptosystems based on logarithmic signatures (also known as group bases) for finite permutation groups. In this paper we show that if $G$ is a nontrivial finite group which is not cyclic of order a prime, or the square of a prime, then the round (or encryption) functions of these systems, that are the permutations of $G$ induced by the exact-transversal logarithmic signatures (also known as transversal group bases), generate the full symmetric group on $G$. This answers a question of S. S. Magliveras, D.R. Stinson and Tran van Trung., This is an old paper, published in 2006

We introduce a generalization $\pounds_{d}^{(\alpha)}(X)$ of the finite polylogarithms $\pounds_{d}^{(0)}(X)=\pounds_d(X)=\sum_{k=1}^{p-1}X^k/k^d$, in characteristic $p$, which depends on a parameter $\alpha$. The special case $\pounds_{1}^{(\alpha)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a {\em grading switching} technique for non-associative algebras. Here we extend such generalization to $\pounds_{d}^{(\alpha)}(X)$ in a natural manner, and study some properties satisfied by those polynomials. In particular, we find how the polynomials $\pounds_{d}^{(\alpha)}(X)$ are related to the powers of $\pounds_{1}^{(\alpha)}(X)$ and derive some consequences., Comment: 18 pages

In this paper we characterize the genera of those quotient curves $\mathcal{H}_q/G$ of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_q$ for which $G$ is contained in the maximal subgroup $\mathcal{M}_1$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a self-polar triangle, or $q$ is even and $G$ is contained in the maximal subgroup $\mathcal{M}_2$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated to $\mathcal{H}_q(\mathbb{F}_{q^2})$. In this way several new values for the genus of a maximal curve over a finite field are obtained. Together with what is known in the literature, our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [Bassa-Ma-Xing-Yeo, J. Combin. Theory Ser. A, 2013] when $G$ fixes a point $P \in \mathcal{H}_q(\mathbb{F}_{q^2})$ and $q$ is even, and the open cases in [Montanucci-Zini, Comm. Algebra, 2018] when $G\leq\mathcal{M}_2$ and $q$ is odd.

Motivated by their study of pro-p limit groups, D. H. Kochloukova and P. A. Zalesskii formulated in [15, Remark after Theorem 3.3] a question concerning the minimum number of generators d ⁢ ( N ) {d(N)} of a normal subgroup N of prime index p in a non-abelian limit group G (see Question*). It is shown that the analogous question for the rational rank has an affirmative answer (see Theorem A). From this result one may conclude that the original question of Kochloukova and Zalesskii has an affirmative answer if the abelianization G ab {G^{\mathrm{ab}}} of G is torsion free and d ⁢ ( G ) = d ⁢ ( G ab ) {d(G)=d(G^{\mathrm{ab}})} (see Corollary B), or if G is a special kind of one-relator group (see Theorem D).

Let [math] be an inner form of a general linear group over a nonarchimedean locally compact field of residue characteristic [math] , let [math] be an algebraically closed field of characteristic different from [math] and let [math] be the category of smooth representations of [math] over [math] . In this paper, we prove that a block (indecomposable summand) of [math] is equivalent to a level- [math] block (a block in which every simple object has nonzero invariant vectors for the pro- [math] -radical of a maximal compact open subgroup) of [math] , where [math] is a direct product of groups of the same type of [math] .

An F -essential subgroup is called a pearl if it is either elementary abelian of order p 2 or non-abelian of order p 3 . In this paper we start the investigation of fusion systems containing pearls: we determine a bound for the order of p-groups containing pearls and we classify the saturated fusion systems on p-groups containing pearls and having sectional rank at most 4.

Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we describe the (intertwining) Hecke algebra $\mathscr{H}(G,K^1)$, that is the convolution $\mathbb{Z}$-algebra of functions from $G$ to $\mathbb{Z}$ that are bi-invariant for $K^1$ and whose supports are a finite union of $K^1$-double cosets. We produce a presentation by generators and relations of this algebra. Finally we prove that the level-$0$ subcategory of the category of smooth representations of $G$ over a unitary commutative ring $R$ such that $p\in R^{\times}$ is equivalent to the category of modules over $\mathscr{H}(G,K^1)\otimes_\mathbb{Z} R$., Comment: 17 pages, minor corrections, updated bibliography

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group., Comment: 11 pages; to appear in Pacific Journal of Mathematics

The intersection growth of a group $G$ is the asymptotic behavior of the index of the intersection of all subgroups of $G$ with index at most $n$, and measures the Hausdorff dimension of $G$ in profinite metrics. We study intersection growth in free groups and special linear groups and relate intersection growth to quantifying residual finiteness., Comment: 20 pages, no figures. Revised version. We extend estimates to polycyclic groups and explain why normal intersection growth is a profinite invariant. Theorem 6.1 previously contained an error which has now been fixed

Let n ≥ 2 and let α ∈ Vnbe an element in the Higman-Thompson group Vn. We study the structure of the centralizer of ∈ Vnthrough a careful analysis of the action of (α) on the Cantor set. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks and flow graphs to assist us in our analysis. A consequence of our structure theorem is that element centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in Vn. © European Mathematical Society.

W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(\rho(G))$, where $\rho$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(\rho(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G, +)$, what are the commutative ring structures $(G, +, \cdot)$ such that all automorphism of $G$ as a group are also automorphisms of $G$ as a ring?, Comment: 18 pages. Accepted for publication, Journal of Algebra, March 2017

We solve the twisted conjugacy problem on Thompson's group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut_+(F) are orbit decidable provided a certain conjecture on Thompson's group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak-Fel'shtyn-Goncalves in [4] showing that F has property R_\infty, and which can be extended to show that Thompson's group T also has property R_\infty., Comment: 31 pages, 2 figures; improved exposition

We consider generalisations of Thompson’s group V, denoted by V r ⁢ ( Σ ) ${V_{r}(\Sigma)}$ , which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, V r ⁢ ( Σ ) ${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type F ∞ ${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type F ∞ ${\operatorname{F}_{\infty}}$ .

It is shown in Lehnert and Schweitzer ('The co-word problem for the Higman-Thompson group is context-free', Bull. London Math. Soc. 39 (2007) 235-241) that R. Thompson's group $V$ is a co-context-free ($co\mathcal {CF}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal {CF}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal {CF}$ group ${\rm QAut}(\mathcal {T}-{2,c})$, which is a group of particular bijections on the vertices of an infinite binary 2-edge-coloured tree, and he conjectures that ${\rm QAut}(\mathcal {T}-{2,c})$ is a universal $co\mathcal {CF}$ group. We show that ${\rm QAut}(\mathcal {T}-{2,c})$ embeds into $V$, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group $V$. In particular, we classify precisely which Baumslag-Solitar groups embed into $V$.

Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $\mathrm{FP}$ it is possible to define an Euler-Poincar\'e characteristic $\chi(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.

Let G F be the group of points fixed by the Frobenius endomorphism F of a simple algebraic group G defined in characteristic p . In this paper, we show that if p is a good prime then the size of every U F -conjugacy class, U F a p -Sylow subgroup of G F , is a q -power, where q is the level of F . We also determine the existence of power series inducing isomorphisms between U F and its Lie algebra.

In this partly expository paper, we study the set [Formula: see text] of groups of orientation-preserving homeomorphisms of the circle S1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in [Formula: see text]. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T.

A finite group G is called p i -central of height k if every element of order p i of G is contained in the k th -term ζ k (G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω1(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P p is less than or equal to the order of the subgroup Ω1(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl p (G) is p-central of height p − 2, then N G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).

In this paper we prove that there exists no function F (m, p) (where the first argument is an integer and the second a prime) such that, if G is a finite permutation p-group with m orbits, each of size at least pF (m,p), then G contains a fixed-point-free element. In particular, this gives an answer to a conjecture of Peter Cameron; see [4], [6].

The aim of this paper is studying the groups in which the zeros of every irreducible character are restricted in being p-elements, for some fixed prime p. In particular, we classify the groups whose irreducible characters vanish only on involutions. Some remarkable examples are also presented. © 2008 Elsevier B.V. All rights reserved.

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups., 8 pages, to appear on the Annales math\'ematiques du Qu\'ebec

It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated, $|G:N

Any morphism $$\phi :\ifmmode\expandafter\hat\else\expandafter\^\fi{G} \to A$$ of profinite groups has maximal l-Frattini quotients $$(\pi ,\beta ),\pi :B \to A,\;\beta :\ifmmode\expandafter\hat\else\expandafter\^\fi{G} \to B,\;{\text{i}}{\text{.e}}{\text{., }}\phi = \pi \circ \beta, \pi$$ is an l-Frattini extension and β is a surjective morphism of profinite groups for which every minimal finite non-trivial l-embedding problem is not weakly solvable. In this paper the case is studied where Ĝ Ĝ is a weakly-orientable l-Poincare duality group of dimension 2 and where A is a finite group whose order is divisible by l. This analysis can be applied for the study of modular towers (Theorem A, Remark 1.2). It is shown that the existence of finite maximal l-Frattini quotients is controlled by an integer r l(A) (Theorem B). In the final section we study properties of the morphism ϕ which imply that for every maximal l-Frattini quotient (π, β), the profinite group B itself is a weakly-orientable l-Poincare duality group of dimension 2 (Theorem C).

The main result of this paper is a graph-theoretic necessary and sufficient condition, for a given set of transvections in SL(n, K) (n > 2 and K a finite field of characteristic not 2 or 3), to generate a group isomorphic to SL (m, L), for some m and some subfield L of K.

Let 1 --> N --> G --> G/N --> 1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n := cd(p)(G) < infinity and if the order of H-k(N, F-p) is finite for k := cd(p)(N), the virtual cohomological p-dimension of G/N equals n - k. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.

We present an efficient implementation of the solution to the conjugacy problem in Thompson's group F. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a description of our solution algorithm, including the data structure that represents strand diagrams and supports simplifications.

In this paper we show that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one-dimensional trivial module of a maximal torus. As a consequence, we obtain that the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p^MT(L), where MT(L) denotes the largest dimension of a torus in L. Finally, we prove that in characteristic p>3 the projective cover of the trivial irreducible L-module is only induced from the one-dimensional trivial module of a torus of maximal dimension if L is solvable., 18 pages

A Bloch-Kato pro-p group G is a pro-p group with the property that the F_p-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientation $\theta\colon G\rightarrow \Z_p^\times$ such that G is theta-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G)=cd(G), and its \F_p-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups (see Theorem A). There are certain Galois-theoretical implications, since Bloch-Kato pro-p groups arise naturally as maximal pro-p quotients and pro-p Sylow subgroups of absolute Galois groups (see Corollary 4.9). Finally, we study certain closure operations of the class of Bloch-Kato pro-p groups, connected with the Elementary type conjecture., Comment: 17 pages, to appear on Forum Math

Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n. In this note we show that the intersection growth of some groups may not be a nicely behaved function by showing the following seemingly contradictory results: (a) for any group G the intersection growth function i_G(n) is super linear infinitely often; and (b) for any increasing function f there exists a group G such that i_G below f infinitely often., 4 pages, no figures

We present an efficient implementation of the solution to the conjugacy problem in Thompson's group F, a certain infinite group whose elements are piecewise-linear homeomorphisms of the unit interval. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a comprehensive description of our solution algorithm, including the data structure that stores strand diagrams and methods to simplify them. We prove that our algorithm theoretically achieves a linear time bound in the size of the input, and we present a quadratic time working solution. © 2013 IEEE.

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F, whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable a in F, M is similar to diag(a Id_h, M_1), where M_1 is cyclic and 0, 28 pages

For a set X an unordered relation on X is a family R of subsets of X. If R is such a relation we let G(R) be the group of all permutations on X that preserves R, that is g belongs to G(R) if and only if x in R implies x^{g} in R. We are interested in permutation groups which can be represented as G=G(R) for a suitable unordered relation R on X. When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group different from the Alternating Group and of degree bigger or equal to 11 is a relation groups. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprmitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation. This property is closely connected to the orbit closure of permutation groups. Since relation groups are orbit closed the results here imply that many classes of imprimitive permutation groups are orbit closed.

Each finite simple group other thanU 3(3) can be generated by three of its involutions. In fact, each such group is generated by two elements, of which one is strongly real and the other is an involution.

A group G ≤ GLK(V) is called K-primitive if there exists no non-trivial decomposition of V into a sum of K-spaces which is stabilised by G. We show that if V is a finite vector space and G a K-primitive subgroup of GLK(V) whose order is coprime to |V|, we can bound the order of G by |V|log2(|V|) apart from one exception. Later we use this result to obtain some lower bounds on the number of p–singular elements in terms of the group order and the minimal representation degree.

Let G be a finite group. An element g ∈ G is called a vanishing element of G if there exists an irreducible complex character of G such that x(g) = 0. In this paper we study the vanishing prime graph Γ(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. Among other things we prove that, similarly to what holds for the prime graph of G, the graph Γ(G) has at most six connected components. © 2010 London Mathematical Society.

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of size ~ n^{2/3} or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups., 6 pages, no figures; final version

In this paper we determine the full character table of a certain split extension $H_1(q)\rtimes Sp(2,q)$ of the Heisenberg group $H_1$ by the odd-characteristic symplectic group $Sp(2,q)$., Comment: 9 pages

We investigate properties of finite transitive permutation groups $(G, \Omega)$ in which all proper subgroups of $G$ act intransitively on $\Omega.$ In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups., Comment: 8 pages, no figures, journal paper