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Constructing All Composition Series of a Finite Group Constructing All Composition Series of a Finite Group Alexander Hulpke Department of Mathematics Colorado State University 1874 Campus Delivery Fort Collins, CO, 80523-1874, USA [email protected] ABSTRACT Subgroups that can occur in a composition series are called This paper describes an effective method for enumerating subnormal, thus a list of all subnormal subgroups will be a all composition series of a finite group, possibly up to action by-product of such an enumeration. of a group of automorphisms. By building the series in an For solvable groups, the set of all composition series also ascending way it only requires a very easy case of comple- parameterizes the set of polycyclic presentations, up to a ment computation and can avoid the need to fuse subspace choice of generators of cyclic groups. (A polycyclic presen- chains in vector spaces. tation [11, Section 8.1] is the standard way of representing As a by-product it also enumerates all subnormal sub- solvable groups on a computer.) groups. A naive approach to enumerating composition series would be to determine the maximal normal subgroups of the whole Categories and Subject Descriptors group and then for each such subgroup U calculate U's max- imal normal subgroups in turn, iterating down. If there is a I.1.2 [Symbolic and Algebraic Manipulation]: Alge- group action one would need to form orbits on each level. braic Algorithms This approach will quickly run into problems. For exam- ` 5 ´ 2 ple consider the group G = 3 × GL5(3) × A5, classify- 5 General Terms ing composition series up to conjugacy in G. For N = 3 there are 24 partial composition series that descend from Algorithms G and contain N. The action of GL5(3) then implies that in each case there is one orbit of composition series through Keywords N. The naive approach however would first enumerate max- imal subspaces and then fuse them back into one orbit each Finite Groups; Composition series; Subnormal subgroups; time, causing much redundancy and making this approach Enumeration infeasible. The approach we use instead will only determine the pos- 1. INTRODUCTION sible composition series in each chief factor once, and then A composition series, that is a series of subgroups each combine these in all possible ways to series for the whole normal in the previous such that subsequent factor groups group. Furthermore, the combination step involves calcula- are simple, is one of the basic concepts in group theory. tions that are far easier than the determination of maximal The aim of this paper is to describe an effective process for normal subgroups. enumerating all composition series of a finite group, possibly up to the action of a group of automorphisms. 2. REDUCTION TO CHIEF FACTORS While the Jordan-H¨older theorem states that the collec- We assume globally that G is a finite group, given in a rep- tion of composition factors (with multiplicity) is an invariant resentation that allows us to test membership in subgroups, of the group, groups can have a huge number of composi- compute subgroup orders (and their prime number factoriza- tion series (see the examples in Section 6.1 below) even when tions), as well as compute a chief series for G. This certainly enumerating up to automorphisms. This shows that such an holds for permutation groups and groups given by a PC pre- enumeration is a nontrivial task, producing useful informa- sentation, which are the cases for which the algorithm has tion. been implemented. Using matrix group recognition [1] there is no fundamental obstacle to apply it also to matrix groups, Permission to make digital or hard copies of all or part of this work for personal or though implementation would be harder. classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full cita- This model allows for the test of subgroup membership tion on the first page. Copyrights for components of this work owned by others than and computation of subgroup orders in factor groups of G. ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- In that it is nominally stronger than the model of a black- publish, to post on servers or to redistribute to lists, requires prior specific permission box group (in which it is only possible to test for element and/or a fee. Request permissions from [email protected]. equality) that is sometimes used to describe computations ISSAC’15, July 6–9, 2015, Bath, United Kingdom. in factor groups; however it models the main classes of rep- Copyright c 2015 ACM 978-1-4503-3435-8/15/07 ...$15.00. DOI: http://dx.doi.org/10.1145/2755996.2756642. resentations of finite groups on a computer as used today, G G D1 Di-1 C2 D2 C C1 Dl 3 Di C D 2 l+1 E1 N Cj-1 E2 N 1 C C i Ej-1 j N 2 Ci+1 Ej=Cj+1 C Nj i+2 Ej+1 =Cj+2 Nj+1 〈1〉 〈1〉 Figure 2: Extension to a single step in the factor Figure 1: Relation to normal subgroup. group and avoids technical assumptions (such as the existence of To see how the enumeration problem reduces, assume that discrete logarithm or element order oracles, or the use of we have classified all composition series for N and for G=N. quantum algorithms) that are otherwise sometimes needed (The base case of the reduction is that of a characteristically in the black-box model. simple group, section 5 describes how to find composition series for these.) We identify a series of subgroups G = C0 ≥ C1 ≥ · · · ≥ Also assume that we have chosen a particular composition Cl = h1i with the set fCig, inclusion providing a natural series N = N0 £ N1 £ ··· £ h1i for N, as well as a particular ordering. (We shall assume that indices are always chosen series of subgroups G = D0 £Di £···£Dk = N that induces compatible with inclusion.) a composition series for G=N. Our goal now is to determine Such a series fCig is called a subnormal series if each all composition series fCig of G such that fN \ Cig = fNig subgroup is normal in the previous one, Ci¡Ci−1. It is called and fNCig = fDig. a composition series if furthermore each factor Ci−1=Ci is We shall construct all these series fCig in a process of simple, that is the series is as fine as possible. ascending over the series fDig. In each step we assume that If we consider instead the case that all subgroups are nor- the composition series for Di are known; the base case being mal in G, such a series fCig is called a normal series; if the series fNig for N = Dk. it is maximally refined it is called a chief series. In a chief To describe a single step, assume that we have a particular series the subsequent factors Ci−1=Ci do not have to be sim- composition series fEj g for Di and want to find all series ple, but must be a isomorphic to a direct power of a simple fCj gj for Di−1 such that fCj \Digj = fEj gj . (Figure 2) As group, that is Ci−1=Ci =∼ Ti × · · · × Ti with Ti simple. (We Di−1 £Di, the description above shows that there is exactly say that Ci−1=Ci is characteristically simple.) one step j such that Cj \ Di = Cj+1 \ Di. We therefore have that Cm+1 = Em for m ≥ j. The standard reduction of algorithmic problems in group For m ≤ j we have that Em = Di \ Cm and Cm = theory is to consider a normal subgroup N ¡ G and recurse hEm;Cj i, implying that Em¡Cm and thus Em¡hEm−1;Cmi = to N and G=N, combining the results in the end. Cm−1. Thus in the factor Cm−1=Em, the subgroup Cm=Em We now assume that fCig is a composition series of G and is a normal (as Cm ¡ Cm−1) complement to Em−1=Em. that N ¡ G. We thus can construct all possible series fCig in a suc- A standard exercise in abstract algebra shows that fCi \ cessive calculation of normal complements, in each step m Ng forms a composition series for N. (Note that there will constructing those series for which Cm \ Di = Cm+1 \ Di. be duplication of subgroups, i.e. for some i we have that Ci \ N = Ci+1 \ N, though Ci 6= Ci+1. Considering a series We describe this construction in more detail: as a set eliminates such duplicates.) Similarly, fNCi=Ngi Let C0 = Di−1 and take (this will be the series for j = 0) is a composition series for G=N. As the concatenation of the concatenation C0 = Di−1 £ E0 £ E1 £ ··· £ h1i. Next, the series NCi with the series N \ Ci yields a composition test whether E1 ¡ C0, if not there are no series for j > series for G, we easily see that for each i we have that either 0. Otherwise determine all normal complements C1=E1 to NCi = NCi+1, or N \ Ci = N \ Ci+1. (See Figure 1) E0=E1 in C0=E1. For each such subgroup C1 we get a series C0 £ C1 £ E1 £ ··· £ h1i; all series for j = 1 are obtained A. Thus if H=A is nonsolvable, every element of H must in- this way. duce an inner automorphism which shows that in this case Next, for each of these series, test whether E2 ¡ C1 and CH (A) is not trivial.
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