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.
Deep Dive into Efficient Isomorphism Testing for a Class of Group Extensions.
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.
The group isomorphism problem is the problem of deciding, for two given groups G and H, whether there exists an isomorphism between G and H, i.e. a one-one map preserving the group operation. This is a fundamental problem in computational group theory but little is known about its complexity. It is known that the group isomorphism problem (for groups given by their multiplication tables) reduces to the graph isomorphism problem [13], and thus the group isomorphism problem is in the complexity class N P ∩ coAM (since the graph isomorphism problem is in this class [2]). Miller [16] has developed a general technique to check group isomorphism in time O(n log n+O (1) ), where n denotes the size of the input groups and Lipton, Snyder and Zalcstein [15] have given an algorithm working in O(log 2 n) space. However, no polynomial algorithm is known for the general case of this problem.
Another line of research is the design of algorithms solving the group isomorphism problem for particular classes of groups. For abelian groups polynomial time algorithms follow directly from efficient algorithms for the computation of Smith normal form of integer matrices [11,6]. More efficient methods have been given by Vikas [24] and Kavitha [12] for groups given by their multiplication tables. The current fastest algorithm solving the abelian group isomorphism problem for groups given as black-boxes has been developed by Buchmann and Schmidt [5] and works in time O(n 1/2 (log n) O (1) ). However, as far as nonabelian groups are concerned, very little is known. For solvable groups Arvind and Torán [1] have shown that the group isomorphism problem is in N P ∩ coN P under certain complexity assumptions but, to our knowledge, the only polynomial-time algorithm testing isomorphism of a nontrivial class of nonabelian groups is a result by Garzon and Zalcstein [7], and holds for a very restricted class.
In this work we focus on the complexity of the group isomorphism problem over classes of nonabelian groups. Since for abelian groups the problem can be solved efficiently, we study one of the most natural next targets: cyclic extensions of abelian groups. Loosely speaking such extensions are constructed by taking an abelian group A and adding one element y that, in general, does not commute with the elements in A. More formally the class of groups we consider in this paper, denoted S , is the following. Definition 1.1. Let G be a finite group. We say that G is in the class S if there exists a normal abelian subgroup A in G and an element y ∈ G of order coprime with |A| such that G = A, y .
In technical words G is an extension of an abelian group A by a cyclic group Z m with gcd(|A|, m) = 1. We will say more about mathematical properties of these extensions in Section 2. For now, we mention that this class of groups includes all the abelian groups and many non-abelian groups too. For example, for A = Z 4 3 and m = 4 we will show that there are exactly 9 isomorphism classes in S .
A group can be represented on a computer in different ways. In this paper we use the black-box setting introduced by Babai and Szemerédi [4], which is one of the most general models for handling groups, and particularly convenient to discuss algorithms running in sublinear time. In order to state precisely the running time of our algorithm, we introduce the following definition. The main result of this paper is the following theorem.
Theorem 1.1. There exists a deterministic algorithm checking whether two groups G and H in the class S given as black-box groups are isomorphic and, if this is the case, computing an isomorphism from G to H. Its running time has for upper bound ( √ n + γ) 1+o (1) , where n = min(|G|, |H|) and γ = min(γ(G), γ(H)).
Notice that, for any group G in the class S , the relation γ(G) ≤ |G| holds. Then the complexity of our algorithm has for upper bound n 1+o (1) , and is almost linear in the size of the groups. Another observation is that, if γ = O(n 1/2 ), then the complexity of our algorithm is n 1/2+o (1) and is of the same order as the best known algorithm testing isomorphism of abelian groups [5] in the black-box setting. This case γ = O(n 1/2 ) corresponds to the rather natural problem of testing isomorphism of extensions of a large abelian group by a small cyclic group.
The outline of our algorithm is as follows. Since a group G in the class S may in general be written as the extension of an abelian group A 1 by a cyclic group Z m 1 and as the extension of an abelian group A 2 by a cyclic group Z m 2 with A 1 ∼ = A 2 and m 1 = m 2 , we introduce (in Section 3) the concept of a standard decomposition of G, which is an invariant for the groups in the class S in the sense that two isomorphic groups have similar standard decompositions (but the converse is false). We also show how to compute a standard decomposition of G efficiently. This allows us to consider only the case where H and G are two extensions of the same abelian grou
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
