CYCLIC PERMUTATIONS IN DOUBLY-TRANSITIVE GROUPS

John P. McSorley Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931-1295. USA. jpmcsorl@mtu.edu

INTRODUCTION

Let Ω be a finite set of size n. A cyclic permutation on Ω is a permutation whose cycle decomposition is one cycle of length n. This paper classifies all finite doubly-transitive permutation groups which contain a cyclic permutation. The classification appears in Table 1.

We use (G, Ω) for a finite doubly-transitive permutation group G acting on a finite set Ω. For other notation and definitions see the self-contained article Cameron [\underline{1}]. CLASSIFICATION

(G, Ω) has a unique minimal normal subgroup N = soc(G), which is either elementary abelian or simple.

In the first case suppose (G, Ω) has an elementary abelian regular normal subgroup N of size p^{d}, where d ≥ 1. Let g ∈ G be a cyclic permutation, it has order p^{d}. Now G ≤ AGL(d, p) ≤ GL(d + 1, p). By considering the JCF of g we have p^{d−1} + 1 ≤ d + 1, so d = 1 or p = d = 2. So G contains no cyclic permutations unless d = 1 or p = d = 2. See Table 1, d = 1 corresponds to row a and p = d = 2 to row b.

In the second case, when N is simple, N is known because of the classification of the finite simple groups. Cameron [\underline{1}] tabulates all simple groups, N, which occur as socles of finite doubly-transitive groups.

We have N ≤ G ≤ Aut(N). For each row of the table in [\underline{1}] we will check such G for cyclic permutations: N = A_{n}: Clearly A_{n} contains a cyclic permutation if and only if n is odd. When n ≥ 5 and n is odd, then Aut(A_{n}) ≅ S_{n}. Hence G ≅ A_{n} or S_{n}, see rows c and d of Table 1.

N = PSL(d, q): Here Zsigmondy’s theorem may be used. If G = P SL(2, 8) there is nothing to prove. Consider GL(1, q^{d})◁ΓL(1, q^{d}) ≤ ΓL(d, q). Except for the case that d = 2 and q is a Mersenne prime, let p be a primitive prime divisor of q^{d} −1 and let P be a Sylow p-subgroup of GL(1, q^{d}). We may check that ΓL(1, q^{d}) = N_{ΓL(d,q)}(P) and GL(1, q^{d}) = C_{ΓL(1,q^{d})}(P). Now p does not divide q − 1, so any cyclic permutation must be the image in P ΓL(d, q) of a cyclic subgroup of ΓL(d, q) containing P or a conjugate, and so must be a conjugate of the image of GL(1, q^{d}). Hence such a cyclic permutation must lie in P GL(d, q). Finally, if d = 2 and q is a Mersenne prime, a similar argument can be made with a subgroup P of order 4. Hence, for every d ≥ 2 and prime power q, a group G for which P SL(d, q) ≤ G ≤ P ΓL(d, q) contains a cyclic permutation if and only if P GL(d, q) ≤ G. See row e of Table 1. Thus, we have decided which subgroups of P ΓL(d, q) have cyclic permutations, see p.179 of Feit [\underline{3}].

N = PSU(3, q): Here we use Liebeck, Praeger, and Saxl [\underline{4}] which lists all maximal factorizations of all finite simple groups and their automorphism groups. Let g ∈ G be a cyclic permutation. In this case N is already doubly-transitive and so we need only consider G = N⟨g⟩. If M is any maximal subgroup of G containing g, then G = MG_{α} is a maximal factorization and appears in these lists.

From the lists on p.13 of [\underline{4}] only G = P SU(3, q) for q = 3, 5, and 8 has a maximal factorization. In the first two cases the group A does not contain an element of order q^{3} + 1, so we may exclude them. In the final case, since G = N⟨g⟩ , so G/N is cyclic, and then this case is out by their remark. Hence, P SU(3, q) contains no cyclic permutations.

N = ^{2}B_{2}(q) and ^{2}G_{2}(q): The lists also take care of these two groups.

N = PSp(2d, 2): Here both permutation representations have even degree, hence a cyclic permutation is an odd permutation, but N is complete.

For the remaining cases we refer to the “Atlas of Finite Groups” by Conway, Curtis, Norton, Parker, and Wilson [\underline{2}]. The only groups which contain cyclic permutations are those with prime degree, see the last three rows of Table 1. (See also p.179 of Feit [\underline{3}].)

This completes the examination of the Table in [\underline{1}]. For every finite doubly-transitive group G we have determined whether or not it contains a cyclic permutation, those which do are listed in Table 1.

TABLE 1

G  n  N
a) AGL(1, p), p any prime  p  C_{p}
b) S_{4}  4  C_{2} × C_{2}
c) S_{n}, n ≥ 5  n  A_{n}
d) A_{n}, n odd and ≥ 5  n  A_{n}
e) Any G with P GL(d, q) ≤ G ≤ P ΓL(d, q)  (q^{d} − 1)/(q − 1)  P SL(d, q)

(d, q) ≠ (2, 2), (2, 3), or (2, 4)

f) P SL(2, 11)  11  P SL(2, 11)
g) M_{11}  11  M_{11}
h) M_{23}  23  M_{23}

REMARKS

(i) The groups S_{2} and S_{3} occur in row a as AGL(1, 2) and AGL(1, 3) respectively. (ii) Groups in rows a and b have an elementary abelian socle, groups in rows c − h a non-abelian simple socle. (iii) No two groups from Table 1 are isomorphic except S_{5} from row c and P GL(2, 5) from row e, these two groups have inequivalent representations being of degrees 5 and 6 respectively.

ACKNOWLEDGMENTS The author would like to thank Professor Peter J. Cameron for help and encouragement

in carrying out the research in this paper, and the referee for many helpful comments.

REFERENCES

\underline{1} P.J. Cameron. Finite Permutation Groups and Finite Simple Groups. Bull. London Math. Soc., \underline{13}, (1981), 1 − 22. \underline{2} J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson. Atlas of Finite Groups. Clarendon Press. Oxford. 1985. \underline{3} W. Feit. Some Consequences of the Classification of Finite Simple Groups. The Santa Cruz Conference on Finite Groups. Proc. Symp. Pure Math., \underline{37}, 1980, 175 − 181. \underline{4} M.W. Liebeck, C.E. Praeger, and J. Saxl. The Maximal Factorizations of the Finite Simple Groups and their Automorphism Groups. Memoirs Amer. Math. Soc., \underline{432}, 1990.