Characterization of Families of Rank 3 Permutation Groups by the Subdegrees I

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1 Vol. XXI, Characterization of Families of Rank 3 Permutation Groups by the Subdegrees I By D.G. HIGMAN 1. Introduction. The terminology and notation of [5] for rank 3 permutation groups are used throughout. We consider two cases of the problem of determining the rank 3 permutation groups for which the degree and subdegrees are specified in terms of a parameter. The results are as follows. Theorem I. I/G is a rank 3 permutation group o/degree m 2 with subdsgrees 1, 2 (m -- 1) and (m -- 1)2, m > 2, then G is isomorphic with a subgroup o/the wreath product Zm ~ Z2 o/the symmetric groups of degrees m and 2, in its usual action on m 2 letters. Theorem H. I/ G is a rank3 permutation group o/ degree (2)with 8ubdegre~8 1, 2(m -- 2) and (m- 2) 2, m > 5, then G is isomorphic with a 4-/old transitive subgroup o/ m in its action on the 2-element subsets, unless one o/the/ollowing holds: (a) G ~ PILL2 (8), (b) # ~ 6 and m ~ 9,17, 27 or 57, (c) /~=7andm=51, or (d) ~u and m , 36, 325, 903 or 8,128. Concerning the exceptional cases in Theorem II, PILL2 (8) as a subgroup of Z~ is the only example of a subgroup of Zm, m ~ 4, which is not 4-fold transitive but has rank 3 on the 2-element subsets (see Lemma 5). The case /~----6, m--" 9 is realized by the group G2 (2) and is known [4] to be the only such group. The remaining cases are undecided. 2. Rank 3 groups and strongly regular graphs. The proofs of Theorems I and II rest on the connection between rank 3 permutation groups of even order and strongly regular graphs [5, 10], and the known characterizations of the graphs of L2-type [9] and triangular type [2, 3] and [6, 7]. The graphs considered in this paper are finite, undirected, and without loops. A graph ~ with n vertices is strongly regular [1] ff there exist integers k, l, ~,/x such that (1) each vertex is adjacent to exactly k vertices and non-adjacent to exactly 1 other vertices, k and l positive, and

2 152 D.G. HIGMAN ARCH. MATH. (2) two adjacent vertices are both joined to exactly 4 other vertices and two nonadjacent vertices are both adjacent to exactly # vertices. Assume that ~r is strongly regular. Then (3) ~Z=k(k--4--2), (4) the minimum polynomial of the adjacency matrix A of fr is (x - 2) (x~ - (4 -/~) x - (2 -/~)), (5) A has k as eigenvalue with multiplicity 2, and ~he multiplicities/, g of the roots r, s of x 2 -- (4 -- #) x -- (k -- #) as eigenvalues of A are respectively ]_ (k + l)s + k and g-- (k + l)r + k 8--F r~8 with ] + g =- k + l, and (6) one of the following holds (a) 2=1,#=4-~-1=2/2 and ]=9=2, or (b) d ----(4 -- #)2 _~ 4(2 -- #) is a square. The strongly regular graph ~r is connected ff and only if/z > 0, and its comple- ment ~ is connected if and only if # < 2. We say that ~ is primitive if ~ and ~ are connected, so that (7) ~r is primitive if and only if 0 < ft < k. If # , then (identity) w (adjacency) is an equivalence relation on the set of vertices, hence (8) If/~ = 0, then 2 -J- 2 [ n, and ff tt = 2, then 1 -J- 1 [n. If G is a rank 3 permutation group of even order on a finite set X, ] X[ ---- n, and if lj and T' are the nontrivial orbits of G in X X, then the graphs ~ = (X, LJ) and = (X, F) are a complementary pair of strongly regular graphs, each admitting G as a rank 3 automorphism ~oup, the parameters k, l and 4,/x being respectively the subdegrees (other than 1) and the intersection numbers of G [5, 10]. Primitivity of the permutation group G is equivalent ~o primitivity of the graph ~. 3. Proof of Theorem I. The wreath product Zm ~ Z2 of the symmetric group Zm of degree m > 2 and the symmetric gtoup Z2 of order 2 is constructed as the split extension of the group N = Zm Zm by X <~>, where ~ acts on iv according to (~l, zt2) ~ = (~2, ~1), gi ~Zm. This group acts as a rank 3 permutation group on X X, X -~ {1, 2... m} according to (i,?')(... ) = (i"~, j=') and (i, ])~ ----(], i), with 2 = 2(m -- 2), 1 = (m -- 2) 9" and # = 2. The action is imprimitive for rn = 2 and primitive for m >= 3. The graph ~fm afforded by the suborbit of len~h 2 (m -- 2), in which vertices (a, b) and (c, d) are adjacent if and only ff they are distinct and a = c or b = d, is isomorphic wit h the line graph of the complete bipartite graph on 2m vertices, i.e. the strongly regular graph of L2-type on 2m vertices. Lemma 1. Xm ~ Z2 is the ]ull automorphism group o/~m, m > 2.

3 Vol. XXI, 1970 Characterization of Families of Permutation Groups I 153 Proof. Each vertex (a, b) is contained in exactly two maximal cliques, with vertices {(a, x)[x = 1, 2..., m} and {(y, b)[y -~ 1, 2,..., m}, respectively. If a is an automorphism of ~m, we have two possibilities. Case 1. (1, 1) ~ = (al, bl) and (1, 2) a = (al, b2). In this case (i, j)a ~ (al,?'~) and (i, 1) ~ ~ (i% bi) with ~, fle Zm such that 1 ~ = ai and 1~ = bl. If i and ] are both =~ 1, then (i, ~), as the unique vertex =~ (1, 1) adjacent to (i, 1) and (i, ~), must be mapped by a onto the unique vertex ~= (ai, bi) adjacent to (i% bi) and (al, ]~), i.e., (i, 2")~ = (i% ~). Hence a = (~, fl) e h r. Case 2. (1, 1)a = (ai, bl) and (1, 2) a = (a2, bi). In this case (1, i) a = (]~, bi) and (2, 1) a -= (al, 2 ~) with (~, fl) ezra, so (i, ])a ---- (i~, i a) ---- (i, ])(~'~)*. Hence a~- (cr ~ ez~ ~Z2. Lemma 2. Let ~ be a strongly regular graph such that k ~ 2(m -- 1) and 1 = (m- 1) 2 ]or some m ~ 2. Then ~ is isomorphic with ~q~m unless m = 4, in which case there is, up to isomorphism, exactly one exceptional graph. Proof. In view of [9], it suffices to prove that/~ = 2. By (8) of Section 2 we have # > 0. By (3) of Section 2 we have #(m-- I) = 2(2m ). Assume first that /~ is even, /i ~ 2#0. Then /~o(m -- i) ~- 2m , so that (#o -- 2)m ~uo We have 2>=0, and since 0</~k=2(m-- 1), 0</zo~m--l. Hence (#o -- 2)m --< /zo < m -- 4, sothat(/io--3)m~--4 and hence #o ----< 2. If/~o = 2, then ~ = -- 1 which is impossible, hence # and /l as claimed. Now assume that/~ is odd, so that 2 [ m -- 1 and we can write m = 2 t + 1, t _>-- 1. Then/~t = 4t ~, that is, ~ = (4 --/z)t -- 1, so that/z ~ 3. By (6) of Section 2 we have that 9t t or t 2 ~- 8t + 4 must be a square according as # = 1 or/~ But the non-negative integral }olutions of the equations y2 _-- 9t 2 + 4t and y2 = t 2.~_ 8t -}- 4 are respectively y , t and y , t ~ 0, contrary to t _--> 1. Since it is known [4] that the exceptional graph in case m = 4 of Lemma 2 does not admit a rank 3 automorphism group, we have Theorem I as an immediate Corollary to Lemmas 1 and 2. If H ~ Zm, m --_> 2, is doubly transitive, then H ~27u is a rank 3 subgroup of Zm ~Z2 on the m 2 letters. There are rank 3 subgroups of Zm l Z2 not of this type and we make no attempt here to classify them. Of course it should be noted that Lemma 2 is purely a result about graphs with no reference to groups. The same remark applies to Lemma 4 in the next section. 4. Proof of Theorem li. The symmetric group 2:m on X ---- {1, 2... m}, m --> 3, acts as a rank 3 group: on ~the set of ( 2 ) 2-element subsets of X, with subdegrees k ~- 2(m -- 2), 5----(m-2 2), and ~u The action is imprimitive if m and g primitive ff m _-- 5. The graph Jm afforded by the snborbit of length 2 (m -- 2), in

4 154 D.G. HIGMAN ARCH. MATH. which two 2-element subsets S and T are adjacent ff and only ff IS n T[ , is isomorphic with the line graph of the complete graph on m vertices, i.e., the gtaph of oo (;) vertices. Lemma 3. Zm is the [ull automorphism group o/jm, m > 5, while 4 has index 2 in the automorphism group o/j4. Proof. The maximal cliques of Jm containing a vertex {a, b} are of two types. [a]---~ {{a,x}lxex- {a}}, (I) [b] = {{y, b) l Y e Z -- {b)) and (II) {a,b}, {b,z} and {a,z}, z~ {a,b}. If a is an automorphism of Jm and {a, b} a = {c, d}, then, if m > 5, [a]a = [c] or [d]. Here we define a n = c if [a] a = [c]. Then it is easy to verify that zr e m and ~z induces a. In case m there are involutions in the automorphism group of ]4 interchanging the two types (I) and (II) of maximal cliques, e.g., ~ mapping {1, 3} onto {2, 4} and fixing all other vertices. We see that <Z4, cr is the full automorphism group of f14. Lemma 4. Let ~ be a strongly regular graph with k = 2(m -- 2) and l =(m;2), " " m > 4. Then ~t -~ 4 unless /x=6 and m=7,9,17,27 or 57, #=7 and m=51, or /.t = 8 and m=28,36,325,903 or 8,128. Proof. If ~u = 0, then m by (8) Section 2 and hence ), by (3), which is impossible by (6). Hence # > 0. By (3) we have (*) #(m-- 3) (2m ). We consider two cases according as/~ is even or odd. Case 1. #----2/xo. In this case /xo(m--3)=2(2m--5--~), = (4 --/x0)m + 3/~ Hence, since /~o < (m -- 2), we have so that 0_--<2)~ (~o-- 4)m < 3//o-- 10 ~ 3(m-- 2) < 3 m m Hence (/~o -- 7) m =< --16 and ~uo < 7. If #0:5or6, then 2-- ~ or4--mso that m = 5 or 4 respectively. In the first case, /x0 = 5 > m -- 2 = 3 and in the second/xo = 6 > m -- 2 = 2, so both are ruled out. The remaining possibilities are: 2 3m--7 9 m m -f m--1 m--2 2 (m -- 2) ~- By (6) of Section 2, the entries in the last row are squares. 8m+l

5 u XXI, 1970 Characterization of Families of Permutation Groups I 155 For /x we have the equation 4y 2 = 9m m -~ 25, for which the (nonnegative integtal) solutions are y , m and y , m = 1, both of which are impossible. m--5 If/x we have by (4) and (5) of Section 2 that r if-- and s = -- 4, so that ] _ 4 mm - 2). Since (m + 3, m) 13 and (m -~ 3, m -- 2) 15, we have that m ~- 3 ] giving m , 9, 12, 17, 27, or 57. If m = 12, then , so this case is impossible. If # , we have the equation y2 ---_ 8ra ~- 1. Writing y w + 1, we have (w+,) m----, k=w 2~-w-4 and l-----~(w 2-~w-6)k, and since 2=1 and /~ ~-- 8, we get r = w -- 3 and s = -- w -- 4 by (4) of Section 2. Hence by (5) of Section 2, f = w(w + 2) (w + 3) (w2 w - 4) S(2w + 1) and since w _= 3 and (2w + 1, w + 2) 13, (2w 1, w + 3) 15 and (2w + 1, w 2 + w-- 4)]17, we must have 2w ~ , giving i.e., w = 7, 8, 25, 42 or 127, m , 36, 325, 903 or 8,128. Case 2. # -= 1 (2). In this case (*) implies that 41m--3, and we write m=4t-~3. Then by (*), ~ ---- (8 --/~)t ~ 1 and ~u ~ 9. The possibilities are: ] t-~l 5t~l 3t~-i t~-i 0 d 49 t2 -~ 32t-~- 4 25t2~12t 9t2 ~ 8t-}- 4 t2 -}- 20t ~ Again, the entries in the last row are squares by (6) of Section 2, so/~ :~ 9. The nonnegative integral solutions of the equations y2 _-- 49t 2 -b 32t ~- 4, y2 _-- 25t 2 ~ 12t and y2 = 9t2 ~_ 8t + 4 are y = 2, t = 0, y---- t = 0 and y = 2, t = 0 respectively, so these cases are ruled out. Finally, the equation y2 -- t2 ~_ 20t ~- 16 has just the solutions y , t = 0, which is ruled out, and y = 20, t = 12, giving m ~ 5t. Lemma 5. I/ G ~ m, m ~ 4, has rank 3 on the 2-element subsets, then G is 4-/old transitive unless m = 9 and G ~ PI~L2 (8). nkl Proof. The cases m and 5 follow from the fact that (b-~ divides I G]. Assume that m > 5 and that G is not 4-fold transitive. Clearly G is transitive on the set of 4-element subsets, i.e., G is 4-homogeneous, and ff S is a 2-element subset, then Gs is transitive on the 4-element sets containing S. Hence ff A is a 4-element set, GA: GA, S ~-- 6. Hence GAlA has order 12, and we have by KA~TO~'S result [8] that possibilities are m G 6 PSL2(5), PFL2(5) 9 PSL2 (8), PFL2 (8) 33 PI'L2 (32). nkl The divisibility ~ [G[ rules out au but the case m = 9, G ---- PFL2 (8).

6 156 D.G. HIGMAN ARCH. MATH. Note. The result stated by KA~TO~ in [8] is proved there only under the additional assumption that GA[ A has order ~: 8, which is quite sufficient for our purposes. According to [2, 3] and [6, 7] if ~ is a strongly regular graph with ]c (m -- 2), 2 and/z ~- 4, then ~ ~ Jm unless m , in which case there are exactly three exceptions. Since none of these exceptional graphs admits a rank 3 automorphism group [4], and since the complement of ~r in the case/z , m has the parameters of J7, Theorem II follows by Lemmas 3, 4 and 5. We remark that for m ~ 4, any rank 3 permutation group with subdegrees 1, 4, 1 is either isomorphic with Z4 or the full automorphism group of J4. The exceptional case ~u , m ~ 9 is realized by the group G2 (2). It is in fact known that there is just one strongly regular graph with these parameters which admits a rank 3 automorphism group, and that G~ (2) is its full automorphism group, no proper subgroup of which has rank 3 [4]. The question of the existence (even of graphs) for the remaining exceptional eases is undecided. References [1] R. C. BosE, Strongly regular gtaphs, partial geometries and partially balanced designs. Pacific g. Math. 13, (1963). [2] L. C. Cm~G, The uniqueness and nonuniqueness of the triangular association scheme. Sci. Record 3, (1959). [3] L. C. C~A:SG, Association schemes of partially balanced block desigus with parameters v , nl = 12, n2 = 15, P~I Sci. Record 4, (1960). [4] M. HEST~S, D. G. HIG~A~ and C. C. Sn~s, Rank 3 permutation groups of small degree (in preparation). [5] D. G. HIG~A~, Finite permutation groups of rank 3. Math. Z. 86, (1964). [6] A. J. HO~'FV~Ze, On the uniqueness of the triangular association scheme. Ann. Math. Star. 31, (1960). [7] A. J. HOFFMA~, On the exceptional case in a characterization of the arcs of a complete graph. IBM J. Res. Develop. 4, (1961). [8] W. M. KANTOR, 4-homogeneous groups. Math. Z. 103, (1968). [9] S. S. Sm~IKEA~DE, The uniqueness of the L2-association scheme. Ann. Math. Star. 30, (1959). [10] C. C. Sn~s, On graphs with rank 3 automorphism groups. J. Comb. Theory (to appear). Anschrift des Autors: D. G. Higman Department of Mathematics University of Michigan Ann Arbor, Mich , USA Eingegangen am