computing normalizers in permutation groups

JOURNAL OF ALGORITHMS4, 163-175 (1983)

Computing Normalizers in Permutation Groups*

GREGORY BUTLER

Department of Computer Science, University of Sydney, Sydney, Australia

Received March 22, 1982

1. INTRODUCTION

This paper continues the description of algorithms [ 1, 3, 4, 5, 9-121 for investigating the structure of a particular permutation group. The role that normalizers play in determining the structure of a group not only makes their computation important in its own right, but also indicates that such computations may become a routine part of more complex algorithms, in the same way that centralizer computations are incorporated into algorithms which compute Sylow subgroups [5] and conjugacy classes of elements [lo].

The bulk of the paper presents two algorithms. The first computes the normalizer of a subgroup Fin a permutation group G. The second is closely related to the first and it determines whether two subgroups are conjugate in G. Several special cases are also discussed. These include the case where F is cyclic, where the automorphism group of F is known, or where G is the symmetric group. The conclusion comments on the efficiency of the algorithms and mentions directions for possible further study. It is followed by tabulated results of the performance of our implementations of the algorithms. These implementations form part of CAYLEY [7], a group theoretical system developed in Sydney.

All the algorithms presented are backtrack searches of G. Hence our prime concern will be to describe the choice of an appropriate base, and the restrictions placed on the images of the base under a normalizing element. For completeness the next section presents the concepts of [3, 41 and the notation of the backtrack algorithms BKTKS and BKTK of [4]. However, it would still be advantageous to the reader to be familiar with [4].

The normalizer algorithm has been used extensively in the study [2] of Held’s group, mainly to compute in the symplectic group Sp(4,4) extended

*Part of this work forms part of the author’s Ph.D. thesis at the University of Sydney under Dr. J. J. Cannon. It was partially supported by the Australian Research Grants Committee and the National Research Council of Canada.

163 0196-6774/83 $3.00

Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any fom reserved.

164 GREGORY BUTLER

by the field automorphism, and in the centralizer of a central involution. These groups have permutation representations of degree 136 and 16, respectively. The algorithm to determine conjugacy of subgroups was applied, by hand and by machine, during the determination of the transitive groups of small degree [6]. Normalizers in the symmetric group are of interest in the study of transitive extensions.

We now present the notation not explained in the next section: If a group has mi orbits of length n, for i = 1,2,. . . , 1, then the orbit structure is denoted by the partition {n;tl, @,. . . , n;l’}.

2. BACKGROUND

Throughout, G will be a permutation group acting faithfully on the finite set X. Elements of G act on the right, so that if x E X and g E G, then xg denotes the image of x under g.

The orbit xG of x under G is the set of images 9 as g runs over G. The stabilizer G, of x in G is the set of elements of G which fii x. A sequence B = [x,, x2,. . . , xk] of points in X is a base for G if no nonidentity element of G fixes each member of B. When it is understood which base is meant we denote Gx,, xz,, ,x,_, by G(‘) for i = 1,2,. . . , k + 1. Hence, there is a chain of stabilizers

G = G(l) >, Gc2) > . . a > G(ktl) = {identity}.

A subset S of G is a strong generating sef of G relative to B if, for each i = 1,2,..., k, S n Go) is a generating set of G(‘). That is, S contains a generating set of each group in the stabilizer chain.

From now on we will assume that B = [x,, x2,. . . , xk] is a base for G and that a strong generating set relative to B is known. Suppose that X is totally ordered and that x,, x2,. . . , xk are the first k points of X. There are induced (lexicographical) orders on G and’on the set of sequences [ y,, y,, . . . , yk] of points in X. There is a correspondence between these orders in the sense that g < h if and only if Bg < Bh.

A backtrack search of G is a means of locating elements with a certain property P by running through the elements in order. The aim of an efficient backtrack search is to recognize sequences of consecutive elements which cannot have the required property; that is, to prune the search. The techniques by which this is done are fully explained in [4]. Here we will outline the concepts and notation necessary for the understanding of the following sections.

A sequence 7’ = [ y,, y2,. . . , y, _ ,I, 1 < r < k + 1, of points is called a partial (base) image. It is complete if r = k + 1, otherwise it is incomplete.

NORMALIZERS IN PERMUTATION GROUPS 165

Suppose T is an incomplete partial image. The set of points which “ usefully” extend T in the search for elements with property P is

X,( T ) = { y E XI 3g E G with property P such that

[ Xl, x 2 )..., x,18 = [Yl, X,...Y,-1, Yb

If the elements with property P form a subgroup H then we also denote X,(T) by X,(T). In particular, X,(T) = (x,““‘)s, for any element g of G which maps [x,, x2,. . . , x,- ,] to T. It is seldom that the set X,(T) (or X,(T)) is known, so the backtrack searches use an easily computed approximation xp( T) (or g,(T)) and insist that X,(T) c x;,(T) c X,(T).

A backtrack search for a single element with a given property P is performed by algorithm BKTKS [4]. The algorithm is completely determined by specifying the property P; the determination of Fp(T) from P and T; how the choice of base B is related to P; and a subgroup K of G with the property that if g E G has property P, then all elements of gK have property P.

If the set of elements with a given property forms a subgroup H then algorithm BKTK [4] computes a strong generating set of H relative to B. If T= [Y,, y,,..., y, _ ,] is the partial image being considered by the algorithm, then s is the greatest integer such that xi = yi for i = 1,2,. . . , s - 1, and K is the subgroup of H which has already been computed. The algorithm is completely determined by specifying the property P; the determination of FH(T) from P and T; how the choice of base is related to P; a subgroup L of H for which a set of generators is known; and which

subgroup of KY,. y2.. y,- , is used to approximate it when r > s. We remark that algorithms for computing a base and strong generating

set are known [9, 121, as are algorithms for changing base [ 111.

3. NORMALIZER

The development of the normalizer algorithm has gone through several phases, from an unadorned backtrack search, which in effect searched the coset representatives of the subgroup, to one which used a knowledge of orbits and their fusion in order to restrict the choice of base images. In this latter approach the base was originally chosen to be compatible (in the sense of [4, Sect. 5.41) with the orbits of the subgroup. The base is now chosen by the more complicated procedure described below which bases its choice on the fusion of the orbits of the groups which form the stabilizer chain of the subgroup.

166 GREGORY BUTLER

Let F be a subgroup of G and suppose that a strong generating set of F relative to B is known. In order to compute the normalizer H of Fin G, we use algorithm BKTK to search for elements with the property

P: “g normalizes F “.

A normalizing element must permute the orbits of F, and furthermore, if the element fixes a point x, then it must also permute the orbits of Fx. The restrictions placed on the base images during the search are derived from these facts.

Let T = IY,, y,,. . . , y, _ ,] be an incomplete partial image and suppose that y, =xiforalli<s.Letj<randtds.Theimagey,ofx,underan element of N,(F) which maps [x1, x2,. . . , x,-i] to [y,, y2,. . . , y,-,], if one exists, is restricted by the following conditions. If x, and xi are in the same orbit of F (‘) then the image y, of x, must lie in yj’(‘). Similarly, if x, is not in the F(‘)-orbit containing xi, then y, cannot lie in y,‘(‘). So we define the condition

x r E x?‘) J

forsomej < randt $ s, (*I

where the largest possible j and t are chosen, and we define the sets

I-(T) = YjF(') if (*>,

=x otherwise,

and

r-l

A(T) = U yi” if(*)andt=s, i-j+ 1

r-1

= i y+, Yr u YjF(‘+‘) if(*)andt<s,

r-1 = y, Yi’ otherwise.

If an element normalizes F and fixes x,, x2,. . . , x,- ,, then it maps the orbits of F(l), F(‘), . . . , F(“) to ones of the same length. Hence define

Z(T) = A {y E XllyF”‘I = lx:(‘)l} if (*), i==I

= ib, {Y E XIIYF”‘I = IxP(“I} otherwise.


The approximation to X,(T) used by algorithm BKTK is

&(T) = X,(T) n (I-‘(T) \ A(T)) n Z(T).

For i = 1,2,. . . , k, the following information about the orbits of F(‘) is computed at the outset and stored:

(a) the length, and a representative of each orbit,

(b) the orbits as a union of linked lists (in order to easily compute the sets r(T), A(T), and Z(T)),

(c) if i > 1, the fusion of the orbits in F(’ - ‘),

(d) which orbits contain the base points, and their lengths, and

(e) which pairs of base points are contained in the same orbit.

Note that if F(‘) = Fci+‘), then the information is not duplicated. Also, as s decreases during the execution of algorithm BKTK the information about FcS+‘) is deleted because it is no longer required.

The choice of base is made with three objectives in mind:

(1) maximize the number of occurrences of x, E xrz, I’,

(2) minimize the number of F(‘)-orbits of length I$“‘(, and

(3) minimize the number of points in F(‘)-orbits of length lxp”‘)I.

If the F(‘)-orbits contained in xFT-” r , are not all trivial, then we qualify (2) and (3) to consider only those orbits contained in x,“lr,” \ {x,, x2,. . . , x,-J

The purpose of (1) is to minimize the size of I’(T) while the purpose of (3) is to minimize lZ(T)(. Let K be the subgroup of H already computed by algorithm BKTK. Then F(“) d K and y, can be chosen from among the first points of the K-orbits of length lxf”‘J. So while (2) plays a role in minimizing [Z(r)/ its primary purpose is to minimize the number of choices of yr when r = s.

We give an example to illustrate the choice of base. The Chevalley group G = G,(4) of degree 416 has a subgroup F isomorphic to L2( 13) with orbits (78, 912, 156). If x, is chosen from the F-orbit of length 78, then the F-orbits split into F,,-orbits as follows: (1, 75, 143}, {73, l45}, {77, l43), and (2, 14i’). So x2 is chosen from an F,,-orbit of length 14 contained in xf: Then F X,,X2 = {identity} and the remaining base points x3, x4,. . . , x7 can be chosen from x$. Hence, there is just one choice for yr, 3 choices for y2, and, for i > 2, yi must be chosen from r2Fx,.

The procedure for choosing the base point x, is: Let Y = xr?; ‘) \ {x*,x2,..., x, _ ,}. Our initial aim is to choose x, E Y, so we consider only the F(‘)-orbits on Y and choose an orbit length m such that the number of

168 GREGORY BUTLER

orbits of length m is minimal. If more than one length occur with minimal frequency, then we choose the smallest length. If m * 1, then x, is chosen to be any point in an orbit of length m. If m = 1, then we choose a point x, which is fixed by F(‘) but not by G (‘). If no such point exists, then we choose the next best value of m and try again. If Go) fixes every point of Y, then we take Y = X \ (CC:?;” U (x,, x2,. . . , x,-i)) and repeat the procedure.

We finish our description of the algorithm by remarking that the known subgroup L is F, and that when r > s then KY,, y2., y, _, is approximated by the trivial subgroup.

4. CONJUGACY OF SUBGROUPS

Let E and F be subgroups of G, and suppose that strong generating sets of E and F relative to B are known. In order to determine whether E and F are conjugate in G, we use algorithm BKTKS [4] to search for an element with the property

P : “g conjugates E to F.".

If 1y is any subgroup of NC(F), for example F, then K satisfies Proposition 2.2 of [4]. Hence we need only consider one element g from each left coset of K.

The restrictions placed on an element g are analogous to those of the normalizer algorithm since g must map orbits of E to corresponding orbits of F.ThebaseB = [x1,x2 ,..., XJ for G is chosen as if we were computing the normalizer of E, and if T = [y,, y2,. . . , y,- ,] is an incomplete partial image, then we define

xp(T)= X,(T)n(I'(T)\ A(T)) n X(T)

as the set of possible images of x,. Here we define the condition

x, E x,““’ for some j -C r and t B s (**)

with j and t as large as possible. Then the sets are defined as follows:

r(T) = ypi..y .)I-, if (* *),

=x otherwise,


r-l A(T) = u Y, if(**)andt=r,

i=j+ 1

r-l

= +)J+ I yi” u vi”yr “2. ..yr if(**)andt<r,

r-l

= y,Y” otherwise,

and

y E X/ly%.+ ,y,-t (=I I} xg”) if (* *),

y E XJlyG,.vz . Y,-, 1-l I}

- x,E”’ otherwise.

If EC”) and F _y,,yz,,, ,JI,-l do not have the same number of orbits of each length, then X,(T) is defined to be the empty set.

Note that the computation of the sets l?(T), A(T), and Z(T) requires [Yl, h.‘.,A-I,... ] to be the base for F. The base for K is also changed to be [Y,, Y~,...,JJ-~,... I.

For i = 1,2,..., k, the following information about the orbits of E(‘) is computed at the outset and stored:

(a) the lengths of the orbits and the number of orbits of each length,

(b) the length of the orbits containing the base points, and

(c) the pairs of base points which are contained in the same orbit.

As each pointy, is chosen, the following information about the orbits of F y,, y2,, ,y, is computed and stored (it is deleted when r decreases, of course):

(d) the length and a representative of each orbit .’ (e) the orbits as a union of linked lists, and

(f) the fusion of the orbits in Fy,,Y, ,.,,, Yr-,, if r > 1.

Clearly the information is not duplicated if y, is a fixed point of Fy,, y2,, ye- ,; that is 2 if E(‘+‘) = II(‘).

Suppose, for example that G is L, (2) of degree 3 1. There is a subgroup E of L,(2) isomorphic to A,, and a base [x,, x2,. . . , x5] of L,(2) such that the lengths of the orbits of E and their fusion are

E :{l, 15 1%

Et3)* (1 m * 3 9 > > -7 1,698,

EC')- (1 * > 1, 0,42, 1, n, 49.

170 GREGORY BUTLER

Furthermore, EC’) = Et*), Ec4) = EC’), x3 E xf(*), and x4, x5 E xf(‘). Sup- pose F is a subgroup of G with the same orbit structure. Then there are at most six complete base images to consider in order to determine whether E and F are conjugate.

The performance of this algorithm when K is taken to be F is given in Table II of the Appendix.

5. SPECIAL CASES OF NORMALIZER ALGORITHM'

If F = (f ) is a cyclic group, then the normalizer of F is generated by the centralizer of f and elements which conjugate f to suitable powers of f. These elements may be rapidly computed by the algorithm [4] which determines whether two elements are conjugate.

Consider the more general case where (I is an automorphism of F whose action on F is known, In the same way that the algorithm for determining whether two elements are conjugate is derived from the centralizer of an element algorithm, an algorithm for determining an element g of G acting on F as u acts can be derived from the centralizer of a subgroup algorithm. This follows from the fact that, if x, = xf-, for some f E F, then xf = (x,“- ,>f’.

Suppose that G is the symmetric group on X = {1,2,. . . , n}, and B = [LZ..., n - 11. Further suppose that F is (m - l)-transitive where m > 2. The method outlined will use the fact that an element normalizing F must permute the F-orbits on X” = {( y,, y,, . . . , Ym)]Yi E X).

Let T, = (iI, i,,. . . , i,) and T2 = (j,, j2,. . . , j,- ,), where all entries are less than r. Define

A(‘j”,, T,) = i$e’F(m)h2,

where h,, h, E F, h, maps [1,2 ,..., m - l] to [i,, i, ,..., i,-,I, and h, maps [1,2,..., m - 11 to [j,, j2,. . . , j, _ ,I. Then A(T,, T2) is precisely the set of points x such that T, and (j,, j,, . . . , j, _ ,, x) are in the same orbit of F. Suppose r E A(T,, T2). If g normalizes F, then

rg E A(T,, T2)g = A(Tf, Tj),

which is determined by the action of g on [ 1,2,. . . , r - 11. In terms of the backtrack algorithm, if T = [ y,, y2,. . . , y, _ ,] is an incomplete partial image, then

X,(T) G A((Yi,, Yi2~***~Y~m)~(Yj,, Y,2y*.*pYj,,_,)).

If several such pairs of tuples T, and T2 are known, then X,(T) must be contained in the intersection of the corresponding sets.

‘The ideas of this section are due to C. C. Sims


The algorithm in this special case consists of randomly selecting sets a m+l, a m+2,. . . , CY, _ , of pairs of tuples. If (T,, T2) E a,, then the entries of T, and T2 are less than r, and r E A(T,, T2). We select several pairs of tuples for each value of r and attempt to minimize the size of

r, = n ACT,, T,). CT,. WEa,

The fact that I’, = {r} implies that r is redundant in the base of No(F). For the backtrack search we define

where T = b,, y,,. . . ,yr-,l, T, = (il, i,,. . . , i,,,), and T2 = (j,, &,. . . , j, _ ,). Furthermore, if X,(T) does not have size II,1 then we can set X,(T) to be the empty set.

Some results for this approach are given in Table III of the Appendix.

6. CONCLUSION

Our experience with the implementations of the general algorithms indi- cate that they are reasonably efficient and can be applied to groups of degree less than 500.’ They may be applicable to groups of higher degree. However, no computations in groups of higher degree have been attempted. Their immediate predecessors, which used an inferior method of choosing the base, were not applicable to groups of higher degree. The special cases, where F is cyclic, or where an automorphism of F is known, are very efficient and certainly applicable to groups of degree 1000 or more.

The algorithm based on the F-orbits on X”, where G is the symmetric group would benefit from an improved method of preprocessing, since the results suggest that most of the time is spent creating the sets I,. At present, the implementation appears restricted to groups of degree about 50; It seems likely that this approach can be generalized to the case where F is intransitive, and to the case where G is not the symmetric group.

The cases in Table I where G is the symmetric group were chosen to illustrate a deficiency of the general approach. The two anomalous cases, M,, in Z,, and Lz(l7) in Z&s, have basic orbits of length 11, 10, 9, 8, and 18, 17, 8, respectively. The last non-trivial stabilizer in the stabilizer chain of L,(17) has orbit partition [l*, 8*]. Both groups provide little distinction between points because of the orbits the points are in. Hence, the general approach is inefficient in these two cases. The case of M,, in Z,, uses the degree twelve representation of M, , which has basic orbits of length 12, 11,

172 GREGORY BUTLER

TABLE I Performance of the Normalizer Algorithm

Base Total change

G 1x1 IGI F IFI MFI time time

M24 24 2’033571123

L,(2) 31 21°3*5’731

G,(4) 416 2” 33 5’7 13

El E2

RI R2

El E2

R, R2

4 AS 4 A5 b(7) LzW A5 46 Ml, Ml, b(3) W7)

::

;:

:: 2’3 5 2’3 5 22 3 5 22 3 5 2237 22 3 7 13

2”3’57 11.51 0.17 2”3 7 7.72 0.66 2’0 32 7 15.55 0.57 2’0 33 5 2.29 0.5 1 2’0 32 5 7 1.18 0.15 2’0 32 5 7 2.38 0.49 2’0 32 7 I .93 0.49 2’0 32 7 2.25 0.56 24 32 52 15.43 2.34 22 3 5 10.24 4.30 22 3 5 13.60 3.52 24 32 5 10.05 3.38 24 32 7 13.86 3.62 22 3 7 .I3 19.23 2.67

1.89 0.20 3.58 0.15

86.18 0.22 2.93 0.44 6.13 0.41

> 500.00

10, 6, and the last stabilizer has orbit partition [13, 3,6]. The general approach is adequate in this case. One solution to the inefficiency of the general approach is outlined in the previous section. Another approach is to realize that an element of N,(F) which fixes a base [x,, x2,. . . , x,] of F must centralize the stabilizer in F of {x,, x2,. . . , x,}. The restrictions of [4, Sect. 5.41 would then remove the anomalies.

The general algorithms should be generalizable to matrix groups by considering the action of the subgroups not on the whole of X but just on the translates of the basic orbits of G. The efficiency will suffer because of this repeated computation of orbits and because of the expense of computing orbits and changing bases in matrix groups.

APPENDIX: TABLES OF PERFORMANCE

The algorithms have been implemented to form part of CAYLEY [7], a group-theoretical system. The following results were obtained on the CDC Cyber 174 of the University of Concordia. All times are in seconds.


TABLE II Performance of the Conjugacy of Subgroups Algorithm

G I4 k IEI k’

‘Total Result time

Time changing

base

z IO IO 9 120 200 200 200 400 720 720 720 720

14400 M24 24 7 4

8 64

L,(2) 31 7 4 16 60 64

168 360

Point stabilizer of w4 72 3 8

4032 SP(4,4) 85 4 4

24 120

26.3X, 96 4 24 G(4) 416 7 24

4 Y 3 Y 3 N 4 N 4 Y 4 N 4 N 3 N 3 Y 7 N 1 Y 1 Y 3 Y 1 Y I N 2 Y 2 Y 3 Y 3 Y

2 Y 3 Y 1 Y 2 Y 3 Y 2 Y 1 Y

0.87 0.31 0.71 0.18 3.37 0.05 0.54 0.27 1.00 0.35 0.54 0.19 0.38 0.04

136.60 0.04 38.43 0.04

1.42 0.56 0.64 0.12 0.49 0.10 1.51 0.64 0.57 0.22 0.33 0.14 1.27 0.53 1.38 0.63

33.34 10.76 2.07 0.96

1.26 0.26 2.04 0.52 7.74 1.88 2.55 1.06 3.39 1.36 3.22 1.20

30.85 5.32

The notation is the same as the section relevant to the algorithm. In addition, in Table II, k’ is the number of non-redundant base levels for E; F has the same order as E, Y means conjugate, N means non-conjugate; and all the listed subgroups of 2,, are transitive.

The notation for various groups follows that of [8]. In particular, Z, is symmetric group of degree n; A, is alternating group of degree n; L,,(q) is the projective special linear group of dimension n over a field of q elements; M, is a Mathieu group; Sp(4,4) is the symplectic group of dimension 4 over GF(4); and E, , E,, R, , and R, are subgroups of a Sylow 2-subgroup of Mz4.

174 GREGORY BUTLER

TABLE III Performance of the Normalizer in the Symmetric Group Algorithm

1x1 F Total Initialization

m JN,(F)/FI time time Ir+,( r = 1,2,...

IO 4 2 2 3.18 2.82 II A6 3 4 3.60 2.90 II M,I 5 1 4.05 3.42 12 Ml, 4 1 5.74 5.36 13 L,(3) 3 1 5.48 5.15 18 L,U? 3 2 12.39 10.05 20 L,(19) 3 2 13.15 II.11 21 L, (4) 3 6 26.21 14.02 24 M24 6 I 195.12 128.71

31

32 L2(31) 3 2 50 L, (49) 3 4

L3W 3 1 213.11 26.11

35.60 26.66 195.81 64.43 97.10 65.07

I23 16 I223 I5 14 I 1s I43 16 1’ 4 1s I2 2 7 23 1'0 12 2 4 2 1’4 13 12 3 4 2 3 42 1’0 i63512615873 6524232 132043’92127 IO 72975261*3 12215 12 2 15 22 1 2 15 2 1’6 12 212 (2 I)3 32 (2 I)3 123 I2 2 23 52 24 3 I3 2 I 2 I5 2 I 2 13 2 13 2 I’6

The author would like to thank Professor Charles Sims for his many helpful discussions.

REFERENCES

I. MICHAEL D. ATKINSON, An algorithm for finding the blocks of a permutation group, Math. Comp. 29 (1975), 911-913.

2. GREGORY BUTLER, The maximal subgroups of the sporadic simple group of Held, J. Algebra 69 (1981), 67-81.

3. GREGORY BUTLER AND JOHN J. CANNON, Computing in permutation and matrix groups I: normal closure, commutator subgroup, series, Math. Comp. 39 (1982) 663-670.

4. GREOORY BUTLER, Computing in permutation and matrix groups II: backtrack algorithm, Math. Comp. 39 (1982) 671-680.

5. GREGORY BUTLER AND JOHN J. CANNON, Computing in permutation and matrix groups III: Sylow subgroups, manuscript.

6. GREGORY BUTLER AND JOHN MCKAY, The transitive groups of degree up to eleven, Comm. of Algebra, in press.

7. JOHN J. CANNON, Software tools for group theory, Proc. A MS Symp. Pure Math. 37 (1980), 495-502.


8. JAMES F. HURLEY AND ARUNAS RUDVALIS, Finite simple groups, Amer. Math. Monthly 84,

9 (1977), 693-714.

9. CHARLES C. SIMS, Computational methods for permutation groups, in “Computational Problems in Abstract Algebra,” Proceedings of conference, Oxford, 1967 (J. Leech, Ed.), pp. 169-183, Pergamon, Oxford, 1970.

IO. CHARLES C. SIMS, Determining the conjugacy classes of a permutation group, in “Com- puters in Algebra and Number Theory,” Proceedings of the Symposium on Applied Mathematics, New York, 1970, (G. Birkhoff and M. Hall, Jr., Eds.), SIAM-AMS Proceed- ings, Vol. 4, pp. 191- 195, Amer. Math. Sot., Providence, RI., 1971.

I I. CHARLES C. SIMS, Computation with permutation groups, in “Proceedings of the Second Symposium on Symbolic and Algebraic Manipulation,” Los Angeles, 1971 (S. R. Petrick, Ed.), pp. 23-28, Association of Computing Machinery, New York, 1971.

12. Jeffrey S. Leon, On an algorithm for finding a base and strong generating set for a group given by generating permutations, Math. Comp. 35, 151 (1980), 941-974.

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