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Representing Groups on Graphs

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Representing Groups on Graphs Representing groups on graphs Sagarmoy Dutta and Piyush P Kurur Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India 208016 {sagarmoy,ppk}@cse.iitk.ac.in Abstract In this paper we formulate and study the problem of representing groups on graphs. We show that with respect to polynomial time turing reducibility, both abelian and solvable group representability are all equivalent to graph isomorphism, even when the group is presented as a permutation group via generators. On the other hand, the representability problem for general groups on trees is equivalent to checking, given a group G and n, whether a nontrivial homomorphism from G to Sn exists. There does not seem to be a polynomial time algorithm for this problem, in spite of the fact that tree isomorphism has polynomial time algorithm. 1 Introduction Representation theory of groups is a vast and successful branch of mathematics with applica- tions ranging from fundamental physics to computer graphics and coding theory [5]. Recently representation theory has seen quite a few applications in computer science as well. In this article, we study some of the questions related to representation of finite groups on graphs. A representation of a group G usually means a linear representation, i.e. a homomorphism from the group G to the group GL (V ) of invertible linear transformations on a vector space V . Notice that GL (V ) is the set of symmetries or automorphisms of the vector space V . In general, by a representation of G on an object X, we mean a homomorphism from G to the automorphism group of X. In this article, we study some computational problems that arise in the representation of finite groups on graphs. Our interest is the following group representability problem: Given a group G and a graph X, decide whether G has a nontrivial representation on X. As expected this problem is closely connected to graph isomorphism: We show, for example, that the graph isomorphism problem reduces to representability of abelian groups. In the other direction we show that even for solvable groups the representability on graphs is decidable using a graph isomorphism oracle. The reductions hold true even when the groups are presented as permutation groups. One might be tempted to conjecture that the problem is equivalent to Graph Isomorphism. However we conjecture that this might not be the case. The non-solvable version of this problem seems to be harder than graph isomorphism. For example, we were able to show that representability of groups on trees, a class of graphs for which isomorphism is decidable in polynomial time, is as hard as checking whether, given an integer n and a group G, the symmetric group Sn has a nontrivial subgroup homomorphic to G, a problem for which no polynomial time algorithm is known. 1 2 Background In this section we review the group theory required for the rest of the article. Any standard text book on group theory, for example the one by Hall [4], will contain the required results. We use the following standard notation: The identity of a group G is denoted by 1. In addition 1 also stands for the singleton group consisting of only the identity. For groups G and H, H ≤ G (or G ≥ H) means that H is a subgroup of G. Similarly by H E G (or G D H) we mean H is a normal subgroup of G. Let G be any group and let x and y be any two elements. By the commutator of x and y, denoted by [x,y], we mean xyx−1y−1. The commutator subgroup of G is the group generated by the set {[x,y]|x,y ∈ G}. We denote the commutator subgroup of G by G′. The following is a well known result in group theory [4, Theorem 9.2.1] Theorem 2.1. The commutator subgroup G′ is a normal subgroup of G and G/G′ is abelian. Further for any normal subgroup N of G such that G/N is abelian, N contains G′ as a subgroup. A group is abelian if it is commutative, i.e. gh = hg for all group elements g and h. A group G is said to be solvable [4, Page 138] if there exists a decreasing chain of groups G = G0 ⊲ G1 . ⊲ Gt = 1 such that Gi+1 is the commutator subgroup of Gi for all 0 ≤ i<t. An important class of groups that play a crucial role in graph isomorphism and related problems are permutation groups. We follow the notation of Wielandt [11] for permutation groups. Let Ω be a finite set. The symmetric group on Ω, denoted by Sym (Ω), is the group of all permutations on the set Ω. By a permutation group on Ω we mean a subgroup of the symmetric group Sym (Ω). For any positive integer n, we will use Sn to denote the symmetric group on {1,...,n}. Let g be a permutation on Ω and let α be an element of Ω. The image of α under g will be denoted by αg. For a permutation group G on Ω, the orbit of α is denoted by αG. Similarly if ∆ be a subset of Ω then ∆g denotes the set {αg|α ∈ ∆}. Any permutation group G on n symbols has a generating set of size at most n. Thus for computational tasks involving permutation groups it is assumed that the group is presented to the algorithm via a small generating set. As a result, by efficient algorithms for permutation groups on n symbols we mean algorithms that take time polynomial in the size of the generating set and n. (i) Let G be a subgroup of Sn and let G denote the subgroup of G that fixes pointwise j ≤ i, (i) g i.e. G = {g|j = j, 1 ≤ j ≤ i}. Let Ci denote a right transversal, i.e. the set of right coset (i) (i−1) representative, for G in G . The ∪iCi is a generating set for G and is called the strong generating set for G. The corner stone for most polynomial time algorithms for permutation group is the Schreier-Sims [9, 10, 3] algorithm for computing the strong generating set of a permutation group G given an arbitrary generating set. Once the strong generating set is computed, many natural problems for permutation groups can be solved efficiently. We give a list of them in the next theorem. Theorem 2.2. Given a generating set for G there are polynomial time algorithms for the following task. 1. Computing the strong generating set. 2. Computing the order of G. By a graph we mean a finite undirected graph. For a graph X, V (X) and E (X) denotes the set of vertices and edges respectively and Aut (X) denotes the group of all automorphisms of X, i.e. permutations on V (X) that maps edges to edges and non-edges to non-edges. 2 Definition 2.3 (Representation). A representation ρ from a group G to a graph X is a homo- morphism from G to the automorphism group Aut (X) of X. Alternatively we say that G acts on (the right) of X via the representation ρ. When ρ is understood, we use ug to denoted uρ(g). A representation ρ is trivial if all the elements of G are mapped to the identity permutation. A representation ρ is said to be faithful if it is an injection as well. Under a faithful action G can be thought of as a subgroup of the automorphism group. We say that G is representable on X if there is a nontrivial representation from G to X. We now define the following natural computational problem. Definition 2.4 (Group representability problem). Given a group G and a graph X decide whether G is representable on X nontrivially. We will look at various restrictions of the above problem. For example, we study the abelian (solvable) group representability problem where our input groups are abelian (solvable). We also study the group representability problem on trees, by which we mean group representability where the input graph is a tree. Depending on how the group is presented to the algorithm, the complexity of the problem changes. One possible way to present G is to present it as a permutation group on m symbols via a generating set. In this case the input size is m +#V (X). On the other hand, we can make the task of the algorithm easier by presenting the group via a multiplication table. In this paper we mostly assume that the group is in fact presented via its multiplication table. Thus polynomial time means polynomial in #G and #V (X). However for solvable representability problem, our results extend to the case when G is a permutation group presented via a set of generators. We now look at the following closely related problem that occurs when we study the repre- sentability of groups on trees. Definition 2.5 (Permutation representability problem). Given a group G and an integer n in unary, check whether there is a homomorphism from G to Sn. Overview of the results Our first result is to show that graph isomorphism reduces to abelian representability problem. In fact we show that graph isomorphism reduces to the representability of prime order cyclic groups on graphs. Next we show that solvable group representability problem reduces to graph isomorphism problem. Thus as far as polynomial time Turing reducibility is concerned abelian group representability and solvable group representability are all equivalent to graph isomor- phism.
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