Related papers: Cayley Linear-Time Computable Groups

We propose a new generalisation of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the…

The group isomorphism problem in computational complexity asks whether two finite groups given by their Cayley tables are isomorphic or not. Although polynomial-time isomorphism tests exist for many specific types of groups, no general…

We show that multiplication can be done in polynomial time on a three counter machine that receives its input as the contents of two counters. The technique is generalized to functions of two variables computable by deterministic Turing…

*by a standard (one-tape) Turing machine. It is well-known that the word problem for hyperbolic groups, whence in particular for free groups, can be solved in linear time. However, these algorithms run on machines more complicated than a…

We construct a polynomial-time algorithm that given a graph $X$ with $4p$ vertices ($p$ is prime), finds (if any) a Cayley representation of $X$ over the group $C_2\times C_2\times C_p$. This result, together with the known similar result…

We show that normalizers and permutational isomorphisms of permutation groups given by generating sets can be computed in time simply exponential in the degree of the groups. The result is obtained by exploiting canonical forms for…

We present a generalization of standard Turing machines based on allowing unusual tapes. We present a set of reasonable constraints on tape geometry and classify all tapes conforming to these constraints. Surprisingly, this generalization…

We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a…

The group isomorphism problem asks whether two finite groups given by their Cayley tables are isomorphic or not. Although there are polynomial-time algorithms for some specific group classes, the best known algorithm for testing isomorphism…

Left-right and conjugation actions on matrix tuples have received considerable attention in theoretical computer science due to their connections with polynomial identity testing, group isomorphism, and tensor isomorphism. In this paper, we…

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $SL_2({\mathbb Z} /p^n {\mathbb Z})$ for odd primes $p$ and $n\geq 2$. For $n=2$, the construction is a complete set of irreducible…

The Cayley table representation of a group uses $\mathcal{O}(n^2)$ words for a group of order $n$ and answers multiplication queries in time $\mathcal{O}(1)$. It is interesting to ask if there is a $o(n^2)$ space representation of groups…

We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies…

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation…

We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ whether two $n\times n$ multiplication tables describe isomorphic…

We present efficient computational solutions to the problems of checking equality, performing multiplication, and computing minimal representatives of elements of free bands. A band is any semigroup satisfying the identity $x ^ 2 \approx x$…

We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group…

For $n>3$, every $n\times n$ partial Cayley matrix with at most $n-1$ holes can be reconstructed by quadrangle criterion. Moreover, the holes can be filled in given order. Without additional assumptions, this is the best possible result.…

For a Baer-local (composition) Fitting formation $\mathfrak{F}$ the polynomial time algorithm for the computation of the $\mathfrak{F}$-radical of a permutation group is suggested. In particular it is showed how one can compute the…

We present efficient algorithms to decide whether two given counting functions on non-abelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on non-abelian free groups…