The Subgroups of PSL(3, q) for odd q

Abstract

(n, q) denote the group of n by n matrices of determinant 1 over the field GF(c7) of q elements ; let PSL(n, q) be equal to SL(n, q) modulo its center. The subgroups of PSL(2, q) were determined by Dickson [12]. Those of PSL(3, q) were determined for odd q by Mitchell [19], using geometric methods. (The results for even q are given by Hartley [18].) In this paper we show that more modern group-theoretic methods can be used for a new determination of the subgroups of PSL(3, q), at least when q is odd. (For a result relevant to of the case of even q, see Suzuki [28].) Our major result is Theorem 1.1. Let q=pa be a power of an odd prime p, and let & be a subgroup ofPSL(3,q) of order > 1. Assume © has no normal elementary-abelian subgroup of order > 1. Then © is isomorphic to one of the following: (1) PSL(3,//);ß|«. (2) PU(3,/>«); 2ß|a. (3) in the case where 3 | (pe-1) and 3ß|a, a group containing the subgroup of type (1) with index 3. (4) in the case where 3 \ (pß + 1) and 6ß\a,a group containing the subgroup of type (2) with index 3. (5) PSLÍ2,//) or PGL(2,/A with ß\a andp^3. (6) PSLÍ2, 5), with q=±l (mod 10). (7) PSL(2, 7), with q3=l (mod 7). (8) A6, A7, or a group containing Ae with index 2, with p = 5 and a even. (9) A6, with q= 1 or 19 (mod 30). Moreover, PSL(3, q) has exactly one subgroup © of each type mentioned (for each indicated value of q, ß), up to conjugacy in GL(3, c7)/Z(SL(3, q)). Here GL(n, q) denotes the group of nonsingular matrices of degree n over the field GF(<7); U(n,q) is the subgroup of SL(n, q2), and U*(n,q) the subgroup of GL(n, q2), consisting of matrices A such that A "1 is the transpose of the matrix obtained from A via the automorphism c-» c" of GF(<72). For any group ®, P@ denotes ®/Z(@) where Z(@) is the center of ©. An, Sn denote the alternating and symmetric groups on n letters. We will give explicit representatives (in matrix form) of all conjugacy classes

Transitive projective planes

“…In this proof and subsequently, we will refer to subgroups of GL(2, q) as being of type y, where y is a number between 1 and 7 corresponding to the list above. When the characteristic is odd, the proof of this result is given in [6,Theorem 3.4]. When the characteristic is even we know that GL(2, q) ∼ = PSL(2, q) × (q − 1).…”

“…Thus for n ≥ 18 we must have m ≤ 4. For n < 18, m ≤ 4 or (n, m) = (14,6). This final case will be dealt with along with other exceptional cases in Section 6.3.9.…”

The 2-transitive permutation representations of the finite Chevalley groups

“…We also note that further information is obtained by replacing q by t in the conclusion of (5.10), and then differentiating at t = 1; this will be done in §7. 6. The degrees of certain characters of the classical groups.…”

Finite groups of fourth-power free order

“…If H is not cyclic, then H satisfies case 6 of [1,Theorem 3.4], then H Q 8 W Z 3 , and thus H D Z l Q 8 W Z 3 since H has order fourth-power free order, as in part (iv).…”