Related papers: Subgroups of minimal index in polynomial time

Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of…

Given a signed permutation on $n$ elements, we need to sort it with the fewest reversals. This is a fundamental algorithmic problem motivated by applications in comparative genomics, as it allows to accurately model rearrangements in small…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $\sigma (G) =\{\sigma_{i} |\sigma_{i}\cap \pi (G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a…

This paper studies the number of monic integer polynomials $f$ of height at most $H$ whose Galois group, endowed with the action on the roots, is isomorphic to a prescribed permutation group $(G,\Omega)$. New upper bounds are obtained for…

In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph…

A finite $p$-group $G$ is said to be $d$-maximal if $d(H)<d(G)$ for every subgroup $H<G$, where $d(G)$ denotes the minimal number of generators of $G$. A similar definition can be formulated when $G$ is acted on by some group $A$. We…

Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios…

This is a report on our long term project to find an algorithm to decide if a finitely presented group has a non-trivial action on a tree.

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is…

In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…

We generalize two results about subgroups of multiplicative group of finite field of prime order. In particular, the lower bound on the cardinality of the set of values of polynomial $P(x,y)$ is obtained under the certain conditions, if…

We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that…

Consider a hypergraph (=set system) $\mathbb{H}$ whose $h$ hyperedges are subsets of a set with w elements. We show that the $R$ minimal hitting sets of $\mathbb{H}$ can be enumerated in polynomial total time $O(Rh^2 w^2)$.

A Cayley graph over a group $G$ is said to be central if its connection set is a normal subset of $G$. We prove that every central Cayley graph over a simple group $G$ has at most two pairwise nonequivalent Cayley representations over $G$…

Let $G$ be a finite abelian group $G$ with $N$ elements. In this paper we give a O(N) time algorithm for computing a basis of $G$. Furthermore, we obtain an algorithm for computing a basis from a generating system of $G$ with $M$ elements…

The $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest…

In this note we give some new results concerning the subgroup commutativity degree of a finite group $G$. These are obtained by considering the minimum of subgroup commutativity degrees of all sections of $G$.

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant…

Let $G=F\ast_\varphi t$ be an HNN extension of a free group $F$ with two equal associated normal subgroups $H_1 = H_2$ of finite index. We prove that the word problem in $G$ is decidable in polynomial time. This result extends to the case…

For a finite non cyclic group $G$, let $\gamma(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$…