On Finite Elation Generalized Quadrangles with Symmetries

We study the structure of finite groups G which act as elation groups on finite generalized quadrangles and contain a full group of symmetries about some line through the base point. Such groups are related to the translation groups of translation transversal designs with parameters depending on those of the quadrangles. Using results on the structure of /^-groups which act as translation groups on transversal designs and results on the index of the Hughes subgroups of finite p-groups, we can show how restricted the structure of elation groups of finite generalized quadrangles with symmetries is. One of our main results is that G is necessarily an elementary abelian 2-group, provided that G has even cardinality. In particular, the elation generalized quadrangle coordinatized by G is a translation generalized quadrangle with G as translation group, that is, G contains full groups of symmetries about every line through the base point. 1. Elation generalized quadrangles and A-gonal families The general background of this paper is the study of finite geometries admitting a particular group of automorphisms such that the geometry under consideration can be coordinatized by that group, and, where the properties of the geometry are reflected in the subgroup structure of the automorphism group. Indeed, in the cases considered here, the existence of such a geometry together with its automorphism group can equivalently be formulated as a combinatorial problem concerning the subgroup structure of the coordinatizing group. Therefore, nonexistence results and examples of these geometric structures can be obtained by considering a pure group theoretic problem. In the present paper, we study the structure of finite groups admitting certain families of subgroups having 'extreme intersection properties' (see the conditions (Kl) and (K2) below). The motivation therefore is due to the fact that these groups arise as automorphism groups of finite elation generalized quadrangles. Since their introduction as generalized polygons by J. Tits in [20], finite generalized quadrangles have been studied intensively in the past three decades. The standard reference, which we refer to for details, is the monograph of S. E. Payne and J. A. Thas [16]. Here we give only the definition of the particular geometry we are interested in, together with its automorphism group. Let s, t ^ 1 be integers. A finite generalized quadrangle J2 of type (s, t) is a triple (£P, Jz?, J), where 0> and Jz? are nonempty and disjoint sets whose elements are called points and lines, respectively, and where J is a subset of the cartesian product & x <£, the incidence relation, satisfying the following conditions. (GQ1) For any Pin 0>, the set 5£P\ = {lzS£:{P,l)eJ} has cardinality / + 1 . (GQ2) For any / in S£, the set 0>l: = {Pe0 :{P,l)eJ} has cardinality s+l. Received 31 May 1994. 1991 Mathematics Subject Classification 51 El2. J. London Math. Soc. (2) 53 (1996) 397-406 398 DIRK HACHENBERGER (GQ3) For any two different points P and Q in 0>, the cardinality of £fP f\ ̂ CQ is at most one. (GQ4) For any two different lines / and m in JSf, the cardinality of 0[ n &m is at most one. (GQ5) For any pair (P, I) in 0> x j£? which is not contained in . / , there exists exactly one pair (Q, m) in J such that (P, m) and (Q, /) are members of J^. A collineation of ^ is a pair (a,/?), where a and /? are permutations of the sets d? and Jzf, respectively, such that the incidence relation is respected, that is, (P,l)eJ if and only if (<*(/*),/?(/)) e . / . It is customary to denote a collineation by a single letter. If 6 is a collineation of i2 which fixes a point P and any line incident with P, then 0 is called a w/ior/ a^ow/ P. A whorl 9 about P is called an elation about P if 9 is the identity on .2 or 0 fixes no point of the set & — P, where P: = {XegP-.&p D !£x ¥" 0} is the set of points which have a line in common with P. A finite generalized quadrangle is called a finite elation generalized quadrangle with elation group G and base point P, and is denoted by (i2, G), if G is a group of elations about P acting sharply transitively on the set & — P. From now on, let (J , G) be a finite elation generalized quadrangle of type (s, t), where s, t > 1. We are interested in the structure of the elation group G, and therefore summarize the fact (due to W. M. Kantor) that G necessarily admits certain families of subgroups having extreme intersection properties (see, for example, [16, Section 8.2]). Let X be any point in & — P. If / is any line in Jzfp then / is not incident with X (by the definition of P) and therefore, by (GQ5), there exists a unique pair (Y,m) in Jf such that (Y,I), and (X,m) are elements of J. Hence there are defined two mappings 0: J£?P -» S£x and cp: <£p -+ P 1 n X, which by (GQ5) are bijections. Now, for any / in ifp, let Al be the stabilizer of the line 0(/) in G, and let Af be the stabilizer of the point (p{l) in G. Then ,4, and ,4f are subgroups of G of order s and st, respectively, and furthermore, At is a subgroup of A*. Moreover, #": = {A^.lG^p} and ^*: = {Af:le£p} are families of / + 1 subgroups of G, satisfying the following conditions: (Kl) AB n C = {1} for pairwise distinct /4, 5, C in J^; (K2) ,4* n 5 = {1} for distinct A, B in P. The pair (J", J"*) is called a 4-gonal family of type (s, t) in G. It was first shown by Kantor (see [11, Theorem 2]) that, conversely, a group of order st admitting a 4-gonal family can be represented as a regular elation group of a suitable generalized quadrangle. A 4-gonal family is therefore also called a Kantor family (see [5, 6]). In [11], it is also shown that many, though not all, of the previously known finite generalized quadrangles are elation generalized quadrangles and therefore constructable via a 4-gonal family in the corresponding elation group. Moreover, in [11], a (at that time) new class of elation generalized quadrangles was constructed. For the discussion of further examples, we refer the reader to the recent survey article [15] of S. E. Payne and to [16, Chapter 10]. Consequently, the existence of finite elation generalized quadrangles is completely settled, if the groups of order st admitting a 4-gonal family together with such a family are known. For this reason, D. Frohardt and X. Chen in [5,6] study the restriction that the existence of a 4-gonal family in a group G imposes on the structure ON FINITE ELATION GENERALIZED QUADRANGLES WITH SYMMETRIES 399 of G. We state two of their main results which are of interest for our considerations (as already mentioned above, we assume that s, t > 1, throughout). THEOREM 1.1. (Frohardt [6]). Let G be a group of order st admitting a 4-gonal family of type (s, t). If t ^ s then s and t are powers of the same prime number p and G is a p-group. THEOREM 1.2. (Chen and Frohardt [5]). Let G be a group of order st admitting a 4-gonal family (3^,3?*) of type (s, t). If there exist two distinct members in 2F which are normal subgroups of G, then s and t are powers of the same prime number p and G is an elementary abelian p-group. In the present paper, we are going to study the structure of groups G admitting a 4-gonal family {3^,3^*) with the additional property that there exists one member of 3F which is a normal subgroup of G. The normality of some member in 3F has the following geometric meaning (see [16, 8.2.2.(iv)]). For a line / in 5£, let I: = {m e &: 0> n 0m ^ 0} be the set of lines which have a point in common with /. Now, in the above situation, At is a normal subgroup of G, if and only if Al is a full group of symmetries about I, that is, any elation in At fixes each line m of I and acts sharply transitively on the set 0m — (0 > lO0 > m) of each such line m. In the next section, we shall see that an elation generalized quadrangle admitting a full group of symmetries about a line through the base point is closely related to some other kind of coset geometries, likewise group constructable, and which also have been intensively studied in the past few years, namely translation transversal designs. Using results on the structure of the translation group of such a design, we draw conclusions on the structure of an elation group with symmetries of a generalized quadrangle. A detailed outline of our results is given subsequently to Theorem 2.1. The following theorem is a short summary. THEOREM 1.3. Let G be a finite group of order st admitting a 4-gonal family (3?,3?*) of type (s, t). If there exists a member A in 3? which is a normal subgroup of G, then s and t are powers of the same prime number p, and necessarily one of the following two cases occurs: (1.3.1) G is elementary abelian ; (1.3.2) p is odd, G/A is nonabelian and has exponent p. 2. On the structure of regular elation groups with symmetries Throughout this section, let (J", #" *) be a 4-gonal family of type (s, t) in a group G, which will be written multiplicatively. We assume that there exists a member A in 3? which is a normal subgroup of G and study the restriction these assumptions impose on the structure of G. First, we consider the set 3?A: = {A*/A}[){BA/A:Be3F-{A}} of subgroups of the factor group G/A. 400 DIRK HACHENBERGER The cardinality of A */A is equal to t and the cardinality of BA/A is equal to s for all B in #" — {A}. Furthermore, as a consequence of (Kl) and (K2), we have that X 0 Y = A/A for any two different members X and Y of 3FA. Moreover, as