cyclic regular subgroups of primitive permutation groups

J. Group Theory 5 (2002), 403–407 Journal of Group Theory( de Gruyter 2002

Cyclic regular subgroups of primitive permutation groups

Gareth A. Jones

(Communicated by A. A. Ivanov)

Abstract. We determine those finite primitive permutation groups which contain a cyclic regu-lar subgroup, thus completing Feit’s description of such groups. To do this, we show that theabelian subgroups of order ðqd � 1Þ=ðq� 1Þ in PGLdðqÞ are, except when d ¼ 2 and q ¼ 3 or 8,just the Singer subgroups.

1 Introduction

Let ðG;WÞ be a finite primitive permutation group containing a cyclic regular sub-group (conditions arising in the classification of polynomials by their monodromygroups; see [10], [14], [15]). Theorems of Burnside [1, §251–252], Galois, Ritt [16] andSchur [17] imply that if G is simply transitive or solvable, then GcAGL1ðpÞ withn ¼ p prime, or G ¼ S4 with n ¼ 4. Feit [8, Theorem 4.1] proved that if G is non-solvable and doubly transitive, then

(a) G ¼ Sn for some nd 5 or G ¼ An for some odd nd 5, or

(b) PSLdðqÞcGcPGLdðqÞ, acting on n ¼ ðqd � 1Þ=ðq� 1Þ points or hyperplanes,

or

(c) G ¼ PSL2ð11Þ, M11 or M23 with n ¼ 11, 11 or 23 respectively.

These groups G are all primitive, and those in (a) and (c) have cyclic regular sub-groups, but this is less clear in (b). The obvious candidates for such subgroups are theSinger subgroups of PGLdðqÞ, conjugates of the group of automorphisms of the pro-jective geometry PGd�1ðqÞ induced by the multiplicative group of GFðqdÞ, acting asa subgroup of GLdðqÞ; see [11], [18]. We will show that with just one exception,where G ¼ PGL2ð8Þ, these are the only cyclic regular subgroups of PGLdðqÞ. Thisextends a result of Feit [7, Lemma 5.1], who showed that for dd 3 the only cyclicsubgroups of order n ¼ ðqd � 1Þ=ðq� 1Þ in PGLdðqÞ are contained in PGLdðqÞ (hisproof needs a slight amendment for PGL3ð4Þ—see our comments in case (ii) of Sec-tion 2). For dd 3, the isomorphism classes of regular subgroups of PGLdðqÞ wereclassified by Ellers and Karzel [6] (see also [4, §1.4.17]), and it follows that any tran-sitive abelian subgroup is cyclic [4, §1.4.18]. We shall extend this to the case dd 2:

Brought to you by | Heinrich Heine Universität DüsseldorfAuthenticated | 134.99.128.41

Download Date | 9/22/13 5:20 PM

Theorem 1. Let A be an abelian subgroup of order n ¼ ðqd � 1Þ=ðq� 1Þ in PGLdðqÞ,where dd 2 and q is any prime power.

(a) If dd 3, or if d ¼ 2 and q0 3 or 8, then A is a Singer subgroup.

(b) In PGL2ð3Þ, with n ¼ 4, A is a Singer subgroup, or a transitive normal Klein four-group contained in PSL2ð3Þ, or one of three conjugate intransitive Klein four-

groups.

(c) In PGL2ð8Þ, with n ¼ 9, A is a Singer subgroup, or one of a conjugacy class ofcyclic regular subgroups not contained in PGL2ð8Þ, or one of a conjugacy class ofintransitive elementary abelian subgroups, also not contained in PGL2ð8Þ.

(In Atlas notation [3], the generators of the Singer subgroups of PGL2ð8Þ lie in con-jugacy classes 9A, 9B and 9C, and those of the other cyclic regular subgroups lie in9D.)

Corollary 2. Let PSLdðqÞcGcPGLdðqÞ. Then G contains a cyclic regular subgroup

if and only if GdPGLdðqÞ. These cyclic regular subgroups form a single conjugacy

class in G consisting of the Singer subgroups of PGLdðqÞ, except in the case G ¼PGL2ð8Þ when G contains a second conjugacy class of such subgroups not contained in

PGL2ð8Þ.

The groups G satisfying PGLdðqÞcGcPGLdðqÞ correspond bijectively to thesubgroups of PGLdðqÞ=PGLdðqÞGGalðGFðqÞ=GFðpÞÞGCe, where q ¼ pe with p

prime. The exceptional nature of PGL2ð8Þ was noted in [9, Lemma 13.8], where thetransitive subgroups of PGL2ðqÞ are determined. With the results stated earlier, Cor-ollary 2 yields

Theorem 3. A primitive permutation group ðG;WÞ of finite degree n has a cyclic regularsubgroup if and only if one of the following holds:

(i) Cp cGcAGL1ðpÞ where n ¼ p is prime;

(ii) G ¼ Sn for some nd 2 or G ¼ An for some odd nd 3;

(iii) PGLdðqÞcGcPGLdðqÞ where n ¼ ðqd � 1Þ=ðq� 1Þ for some dd 2;

(iv) G ¼ PSL2ð11Þ, M11 or M23 where n ¼ 11, 11 or 23 respectively.

2 Proof of Theorem 1

Our argument extends that of Feit in [7, Lemma 5.1]. We put F ¼ GFðqÞ andK ¼ GFðqdÞ. If d and q are understood, we denote PGLdðqÞ and related groups byPGL, etc. Let A be an abelian subgroup of PGL of order n ¼ ðqd � 1Þ=ðq� 1Þ. Ifq ¼ pe where p is prime, then jPGL : PGLj ¼ e, with

Gareth A. Jones404



jPGLj ¼ nqdðd�1Þ=2Yd�1

i¼1

ðqi � 1Þ:

Now we need

Zsigmondy’s Theorem ([19], also [13, (IX.8.3)]). Let a, i be integers greater than 1.Then except when a ¼ 2b � 1 and i ¼ 2, or a ¼ 2 and i ¼ 6, there is a prime r whichdivides ai � 1 but does not divide a j � 1 for any j such that 0 < j < i, and which doesnot divide i.

In our case it follows that there is a prime r which divides pde � 1 ¼ qd � 1, but doesnot divide p j � 1 for j ¼ 1; . . . ; de� 1, and does not divide de, except in the cases

(i) p ¼ 2b � 1 (a Mersenne prime) and de ¼ 2,

(ii) p ¼ 2 and de ¼ 6.

Suppose that we are not in case (i) or (ii). Since r divides n but not jPGLj=n, Acontains a Sylow r-subgroup R of PGL. By [12, (II.7.3(b))], the centralizer CPGLðRÞof R in PGL is a Singer subgroup, and by a straightforward modification of the proof,embedding GL in GLdeðpÞ, so is CPGLðRÞ. Since AcCPGLðRÞ, with jAj ¼ n, A is aSinger subgroup.

In case (i) we have d ¼ 2 and e ¼ 1, so that AcPGL2ðpÞ ¼ PGL2ðpÞ wherep ¼ 2b � 1. Then A has index 2 in a Sylow 2-subgroup T of PGL. Being dihedral, Thas three subgroups of index 2: two dihedral groups and a Singer subgroup. If bd 3only the Singer subgroup is abelian, and the result follows. If b ¼ 2 (so that n ¼ 4 andPGL ¼ S4) the dihedral subgroups are also abelian: one is the transitive normal Kleinfour-group in S4, and the other lies in a conjugacy class of three intransitive Kleinfour-groups, as in Theorem 1(b).

In case (ii), p ¼ 2 and de ¼ 6, so that d ¼ 2, 3 or 6 with e ¼ 3, 2 or 1. If d ¼ 2 ande ¼ 3 then n ¼ 9 and PGL ¼ PGL2ð8Þ is a split extension of PGL by the Galois grouphbiGC3 of GFð8Þ. A Sylow 3-subgroup T ¼ ha; b j a9 ¼ b3 ¼ 1; ab ¼ a4i of PGLhas order 27, with three cyclic regular subgroups: a Singer subgroup hai ¼ T VPGL,and subgroups habi and hab2i conjugate in PGL and not contained in PGL. Theremaining maximal subgroup of T is an intransitive group ha3; biGC3 � C3. ThusPGL2ð8Þ has three conjugacy classes of abelian subgroups of order 9, as claimed inTheorem 1(c).

If d ¼ 3 and e ¼ 2 then PGL ¼ PGL3ð4Þ, with n ¼ 21. In [7], in his proof of Lemma5.1, Feit claims that one can apply the earlier main argument with r ¼ 7 (given byZsigmondy’s Theorem with a ¼ q ¼ 4 and i ¼ d ¼ 3). However, CPGLðRÞGC7 � S3,which has order 42, not 21 (note the elements of order 14 in

G:22 ¼ PSL3ð4ÞcPGL3ð4Þ

in [3]); the argument fails here because r ð¼7Þ divides p j � 1 ð¼23 � 1Þ for somej < de ð¼6Þ. Nevertheless, CPGLðRÞ has a unique abelian subgroup A of order 21,

Cyclic regular subgroups of primitive permutation groups 405



and since the Sylow 7-subgroups R of PGL are all conjugate, so are the correspond-ing groups A; at least one of them is a Singer subgroup, and so they all are.

Finally, if d ¼ 6 and e ¼ 1 then PGL ¼ PGL6ð2Þ ¼ GL6ð2Þ; as in the main argu-ment, any abelian subgroup A of order n ¼ 63 acts irreducibly and is therefore aSinger subgroup.

(Note that the above minor error in the proof of Feit’s Lemma 5.1 is not the errorin his Lemma 3.4 noted by Muller in [15]; our proof is independent of that result.)

3 Proof of Corollary 2

Let PSLcGcPGL. By Theorem 1, apart from the case of PGL2ð8Þ, G containsa cyclic regular subgroup C if and only if GVPGL does, in which case C is aSinger subgroup; it is therefore su‰cient to determine which groups G satisfyingPSLcGcPGL contain Singer subgroups. Since PGL=PSL is abelian, and theSinger subgroups S of PGL are all conjugate, they have the same image S:PSL=PSLin PGL=PSL. As in the proof of [12, (II.7.3(b))], let S be induced by a primitiveroot a A K � cGLdðqÞ, so that the eigenvalues of a are its algebraic conjugatesaq

i ði ¼ 0; . . . ; d � 1Þ, and detðaÞ ¼ an where n ¼ ðqd � 1Þ=ðq� 1Þ. Since a has orderqd � 1, detðaÞ has order q� 1 and is therefore a primitive root in F �. ThusS:PSL ¼ PGL, so the only groups G between PSL and PGL containing cyclic regularsubgroups are those containing PGL. By Theorem 1, the same is true for PGL2ð8Þ,which contains PGL2ð8Þ ¼ PSL2ð8Þ with index 3, though now there are two con-jugacy classes of cyclic regular subgroups, one in PGL2ð8Þ and the other not.

Acknowledgements. The author is grateful to Alexander Zvonkin for some stimulat-ing conversations about this subject, to the Universite de Bordeaux I for financiallysupporting research visits during which this work took place, and to the referee forhelpful comments.

References

[1] W. Burnside. Theory of groups of finite order, 2nd edn. (Cambridge University Press,1911).

[2] P. J. Cameron. Permutation groups and finite simple groups. Bull. London Math. Soc. 13(1981), 1–22.

[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. Atlas of finitegroups (Clarendon Press, 1985).

[4] P. Dembowski. Finite geometries (Springer, 1968).[5] L. E. Dickson. Linear groups, with an exposition of the Galois field theory (Dover, 1958).[6] E. Ellers and H. Karzel. Endliche Inzidenzgruppen. Abh. Math. Sem. Hamburg 27 (1964),

250–264.[7] W. Feit. On symmetric balanced incomplete block designs with doubly transitive auto-

morphism groups. J. Combin. Theory Ser. A 14 (1973), 221–247.[8] W. Feit. Some consequences of the classification of finite simple groups. In The Santa

Cruz conference on finite groups, Proc. Sympos. Pure Math. 37 (American MathematicalSociety, 1980), pp. 175–181.

Gareth A. Jones406



[9] D. A. Foulser. The flag-transitive collineation groups of the finite Desarguesian a‰neplane. Canad. J. Math. 16 (1964), 443–472.

[10] R. M. Guralnick and J. Saxl. Monodromy groups of polynomials. In Groups of Lie typeand their geometries, London Math. Soc. Lecture Note Ser. 207 (Cambridge UniversityPress, 1995), pp. 125–150.

[11] J. W. P. Hirschfeld. Projective geometries over finite fields, 2nd edn. (Clarendon Press,1998).

[12] B. Huppert. Endliche Gruppen, vol. 1 (Springer-Verlag, 1979).[13] B. Huppert and N. Blackburn. Finite groups, vol. 2 (Springer-Verlag, 1982).[14] G. A. Jones and A. Zvonkin. Orbits of braid groups on cacti. Moscow Math. J. 2 (2002),

129–162.[15] P. Muller. Primitive monodromy groups of polynomials. In Recent developments in the

inverse Galois problem, Contemp. Math. 186 (American Mathematical Society, 1995), pp.385–401.

[16] J. F. Ritt. On algebraic functions which can be expressed in terms of radicals. Trans.Amer. Math. Soc. 24 (1922), 21–30, 324.

[17] I. Schur. Zur Theorie der einfach transitiven Permutationsgruppen. S. B. Preuss. Akad.Wiss., Phys.-Math. Kl. (1933), 598–623.

[18] J. Singer. A theorem in finite projective geometry and some applications to numbertheory. Trans. Amer. Math. Soc. 43 (1938), 377–385.

[19] K. Zsigmondy. Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3 (1892), 265–284.

Received 8 May, 2001; revised 4 March, 2002

G. A. Jones, Department of Mathematics, University of Southampton, Southampton SO171BJ, UK.E-mail: [email protected]

Cyclic regular subgroups of primitive permutation groups 407



cyclic regular subgroups of primitive permutation groups

Documents