Isomorphism Problem For Groups Research Articles

On the modular isomorphism problem for groups with center of index at most p3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^3$$\\end{document}

Let p be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite p-groups whose center has index p3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^3$$\\end{document}, which is a strong contrast to the analogous situation for p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p = 2$$\\end{document}.

Algebraic Interpretation of the Weisfeiler–Leman Algorithm: Schur Ring Analysis of Direct Power Groups

In this paper, we explore the algebraic interpretation of the partitioning obtained by the 𝑚-dimensional Weisfeiler–Leman algorithm on the direct power Gm of a finite group G. We define and study a Schur ring over Gm, which provides insights into the structure of the group G. Our analysis reveals that this ring determines the group G up to isomorphism when m≥3. Furthermore, we demonstrate that as m increases, the Schur ring associated with the group of automorphisms of G acting on Gm emerges naturally. Surprisingly, we establish that finding the limit ring is polynomial-time equivalent to solving the group isomorphism problem. This paper presents a novel algebraic framework for understanding the behavior of the Weisfeiler–Leman algorithm and its implications for group theory and computational complexity.

On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center

Abstract We show that the modular isomorphism problem has a positive answer for groups of nilpotency class 2 with cyclic center, i.e., that for such p-groups G and H an isomorphism between the group algebras FG and FH implies an isomorphism of the groups G and H for F the field of p elements. For groups of odd order this implication is also proven for F being any field of characteristic p. For groups of even order we need either to make an additional assumption on the groups or on the field.

On multidimensional Schur rings of finite groups

Abstract For any finite group 𝐺 and a positive integer 𝑚, we define and study a Schur ring over the direct power G m G^{m} , which gives an algebraic interpretation of the partition of G m G^{m} obtained by the 𝑚-dimensional Weisfeiler–Leman algorithm. It is proved that this ring determines the group 𝐺 up to isomorphism if m ≥ 3 m\geq 3 , and approaches the Schur ring associated with the group Aut ⁡ ( G ) \operatorname{Aut}(G) acting on G m G^{m} naturally if 𝑚 increases. It turns out that the problem of finding this limit ring is polynomial-time equivalent to the group isomorphism problem.

The Modular Isomorphism Problem for small groups – revisiting Eick's algorithm

We study the Modular Isomorphism Problem for groups of small order based on an improvement of an algorithm described by Eick. Our improvement allows to determine quotients I(kG)/I(kG)m of the augmentation ideal without first computing the full augmentation ideal I(kG). Our computations yield a positive answer to the MIP for groups of order 37 and strongly reduce the cases that need to be checked for groups of order 56. We also show that the counterexamples to the Modular Isomorphism Problem found recently by García-Lucas, Margolis and del Río are the only 2- or 3-generated counterexamples of order 29. Furthermore, we provide a proof for an observation of Bagiński, which is helpful in eliminating computationally difficult cases.

On the modular isomorphism problem for groups of class 3 and obelisks

Abstract We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.

Nearly Linear Time Isomorphism Algorithms for Some Nonabelian Group Classes

The isomorphism problem for groups, when the groups are given by their Cayley tables is a well-studied problem. This problem has been studied for various restricted classes of groups. Kavitha gave a linear time isomorphism algorithm for abelian groups (JCSS 2007). Although there are isomorphism algorithms for certain nonabelian group classes represented by their Cayley tables, the complexities of those algorithms are usually super-linear. In this paper, we design linear and nearly linear time isomorphism algorithms for some nonabelian groups. More precisely,

Beating the Generator-Enumeration Bound for Solvable-Group Isomorphism

We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the n log p n + O (1) generator-enumeration algorithm where n is the order of the group and p is the smallest prime divisor of n . In previous work with Fabian Wagner, we showed an n (1 / 2) log p n + O (log n / log log n ) -time algorithm for testing isomorphism of p -groups by building graphs with degree bounded by p + O (1) that represent composition series for the groups and applying Luks’ algorithm for testing isomorphism of bounded-degree graphs. In this work, we extend this improvement to the more general class of solvable groups to obtain an n (1 / 2) log p n + O (log n / log log n) -time algorithm. In the case of solvable groups, the composition factors can be large which prevents previous methods from outperforming the generator-enumeration algorithm. Using Hall’s theory of Sylow bases, we define a new object that generalizes the notion of a composition series with small factors but exists even when the composition factors are large. By constructing graphs that represent these objects and running Luks’ algorithm, we obtain our algorithm for solvable-group isomorphism. We also extend our algorithm to compute canonical forms of solvable groups while retaining the same complexity.

Beating the generator-enumeration bound for p-group isomorphism

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G≅H. For several decades, the nlogp⁡n+O(1) generator-enumeration bound (where p is the smallest prime dividing the order of the group) has been the best worst-case result for general groups. In this work, we show an improvement over the generator-enumeration bound for p-groups, which are believed to be the hard case of the group isomorphism problem. We start by giving a Turing reduction from group isomorphism to n(1/2)logp⁡n+O(1) instances of p-group composition-series isomorphism. By showing a Karp reduction from p-group composition-series isomorphism to testing isomorphism of graphs of degree at most p+O(1) and applying algorithms for testing isomorphism of graphs of bounded degree, we obtain an nO(p) time algorithm for p-group composition-series isomorphism. Combining these two results yields an algorithm for p-group isomorphism that takes at most n(1/2)logp⁡n+O(p) time. This algorithm is faster than generator-enumeration when p is small and slower when p is large. Choosing the faster algorithm based on p and n yields an upper bound of n(1/2+o(1))log⁡n for p-group isomorphism.

Groups with Identical k-Profiles

We show that for $1 \le k \le \sqrt{2\log_3 n}-(5/2)$, the multiset of isomorphism types of $k$-generated subgroups does not determine a group of order at most $n$. This answers a question raised by Tim Gowers in connection with the Group Isomorphism problem.

Solvable Group Isomorphism Is (Almost) in NP ∩ coNP

The Group Isomorphism problem consists in deciding whether two input groups G 1 and G 2 given by their multiplication tables are isomorphic. We first give a 2-round Arthur-Merlin protocol for the Group Nonisomorphism problem such that on input groups ( G 1 , G 2 ) of size n , Arthur uses O (log 6 n ) random bits and Merlin uses O (log 2 n ) nondeterministic bits. We derandomize this protocol for the case of solvable groups showing the following two results: (a) We give a uniform NP machine for solvable Group Nonisomorphism, that works correctly on all but 2log O (1)( n ) inputs of any length n . Furthermore, this NP machine is always correct when the input groups are nonisomorphic. The NP machine is obtained by an unconditional derandomization of the aforesaid AM protocol. (b) Under the assumption that EXP \not\subseteq i.o--PSPACE we get a complete derandomization of the aforesaid AM protocol. Thus, EXP \not\subseteq i.o--PSPACE implies that Group Isomorphism for solvable groups is in NP ∩ coNP.

On the Cayley isomorphism problem for a digraph with 24 vertices

In this paper we are mainly concerned with the Cayley isomorphism problem for groups containing $Q_8$. We prove that the group $Q_8\times C_3$ is not a CI-group with respect to colour ternary relational structures. Further, we prove that the non-nilpotent group $Q_8\rtimes C_3$ is not a CI-group with respect to graphs.

AnO(n) Algorithm for Abelianp-Group Isomorphism and anO(n log n) Algorithm for Abelian Group Isomorphism

The isomorphism problem for groups is to determine whether two finite groups are isomorphic. The groups are assumed to be represented by their multiplication tables. Tarjan has shown that this problem can be done in timeO(nlogn+O(1)) for groups of cardinalityn. Savage has claimed an algorithm for showing that if the groups are Abelian, isomorphism can be determined in timeO(n2). We improve this bound and give anO(nlogn) algorithm for Abelian group isomorphism. Further, we show that if the groups are Abelianp-groups then isomorphism can be determined in timeO(n). We also show that the elementary divisor sequence for an Abelian group can be determined in timeO(nlogn) and for an Abelianp-group it can be determined in timeO(n).

Nondeterministic circuits, space complexity and quasigroups

By considering nondeterminism as a quantifiable resource, we introduce new nondeterministic complexity classes obtained from NC circuits using a bounded number of nondeterministic gates. Let NNC( f( n)) denote the class of languages computable by an NC circuit family with O( f( n)) nondeterministic gates. If f( n) is limited to log n, we show that the class obtained is equivalent to NC. If f( n) is allowed to encompass all polynomials, we show that the class obtained is equivalent to NP. The class of most interest, NNC(polylog), obtained by letting f( n) encompass all polylogarithmic functions, contains a version of the quasigroup (Latin square) isomorphism problem. The quasi-group isomorphism problem is not known to be in P or NP-complete; thus, NNC(polylog) is a candidate for separating NC and NP. We also show that NNC(polylog) ⊆ DSPACE(polylog). More specifically, we show that NNC k (log k n) is contained in DSPACE(log k n), where NNC k (log j n) denotes the complexity class obtained from an NC k circuit with O(log j n) nondeterministic gates. This containment yields DSPACE(log 2 n) algorithms for the quasigroup isomorphism problem, the Latin square isotopism problem and the Latin square graph isomorphism problem. The only previously known bound for these problems is Miller's time bound of n log2 n + O(1) . This result also generalizes the DSPACE(log 2 n) algorithm of Lipton (1976) for the group isomorphism problem. We also show that, for every k, NNC( n k ) ⊆ DSPACE( n k ), and if for some k there exists a j such that DSPACE( n k ) ⊆ NNC( n j ) then NP = PSPACE.

The Modular Group Algebras of P-Groups of Maximal Class

The isomorphism problem for modular group algebras of finite p-groups appears to be still far from a solution (see [7] for a survey of the existing results). It is therefore of interest to investigate the problem for special classes of groups.The groups we consider here are the p-groups of maximal class, which were extensively studied by Blackburn [1]. In this paper we solve the modular isomorphism problem for such groups of order not larger than pp+1, having an abelian maximal subgroup, for odd primes p.What we in fact do is to generalize methods used by Passman [5] to solve the isomorphism problem for groups of order p4. In Passman's paper the case of groups of maximal class is actually the most difficult one.

The word problem and the isomorphism problem for groups

The word problem and the isomorphism problem for groups

Isomorphism Testing for Graphs, Semigroups, and Finite Automata are Polynomially Equivalent Problems

Two problems are polynomially equivalent if each is polynomially reducible to the other. The problems of testing either two graphs, two semigroups, or two finite automata for isomorphism are shown to be polynomially equivalent. For graphs the isomorphism problem may be restricted to regular graphs since we show that this is equivalent to the general case. Using the techniques of Hartmanis and Berman we then show that this equivalence is actually a polynomial isomorphism. It is conjectured that the isomorphism problem for groups is not in this equivalence class, but that it is an easier problem. If the conjecture is true then $P \ne NP$; if it is false then there exists a “subexponential” $O(n^{c_1 \log n + c_2 } )$ algorithm for graph isomorphism.