20 research outputs found

    Faster Isomorphism for pp-Groups of Class 2 and Exponent pp

    Full text link
    The group isomorphism problem determines whether two groups, given by their Cayley tables, are isomorphic. For groups with order nn, an algorithm with n(logn+O(1))n^{(\log n + O(1))} running time, attributed to Tarjan, was proposed in the 1970s [Mil78]. Despite the extensive study over the past decades, the current best group isomorphism algorithm has an n(1/4+o(1))lognn^{(1 / 4 + o(1))\log n} running time [Ros13]. The isomorphism testing for pp-groups of (nilpotent) class 2 and exponent pp has been identified as a major barrier to obtaining an no(logn)n^{o(\log n)} time algorithm for the group isomorphism problem. Although the pp-groups of class 2 and exponent pp have much simpler algebraic structures than general groups, the best-known isomorphism testing algorithm for this group class also has an nO(logn)n^{O(\log n)} running time. In this paper, we present an isomorphism testing algorithm for pp-groups of class 2 and exponent pp with running time nO((logn)5/6)n^{O((\log n)^{5/6})} for any prime p>2p > 2. Our result is based on a novel reduction to the skew-symmetric matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop several tools for matrix space analysis, including a matrix space individualization-refinement method and a characterization of the low rank matrix spaces.Comment: Accepted to STOC 202

    Isomorphism in expanding families of indistinguishable groups

    Full text link
    For every odd prime pp and every integer n12n\geq 12 there is a Heisenberg group of order p5n/4+O(1)p^{5n/4+O(1)} that has pn2/24+O(n)p^{n^2/24+O(n)} pairwise nonisomorphic quotients of order pnp^{n}. Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most pp. They are also directly and centrally indecomposable and of the same indecomposability type. The recognized portions of their automorphism groups are isomorphic, represented isomorphically on their abelianizations, and of small index in their full automorphism groups. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.Comment: 28 page

    Efficient Isomorphism Testing for a Class of Group Extensions

    Get PDF
    The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. In this paper we study this problem for a class of groups corresponding to one of the simplest ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group A by a cyclic group of order m. We present an efficient algorithm solving the group isomorphism problem for all the groups of this class such that the order of A is coprime with m. More precisely, our algorithm runs in time almost linear in the orders of the input groups and works in the general setting where the groups are given as black-boxes.Comment: 17 pages, accepted to the STACS 2009 conferenc

    On the isomorphism problem for certain pp-groups

    Full text link
    We consider 9 infinite families of finite pp-groups, for pp a prime, and we settle the isomorphism problem that arises when the parameters that define these groups are modified
    corecore