Examples of highly transitive permutation groups

Transcription

1 Sonderdrucke aus der Albert-Ludwigs-Universität Freiburg OTTO H. KEGEL Examples of highly transitive permutation groups Originalbeitrag erschienen in: Rendiconti del Seminario Matematico della Università di Padova 63 (1980), S

2 REND. SEAS. MAT. UNIV. PADOVA, Vol. 63 (1980) Examples of Highly Transitive Permutation Groups. OTTO H. KEGEL (*) The permutation group (G, D) is called highly transitive if, for every natural number n, the group G acts n-fold transitively on the infinite set Q. Every subgroup of the symmetric group on Q which contains the group of all even finite permutations of S2 trivially provides an example of a highly transitive permutation group. In mainly unpublished discussions over the years other examples of this phenomenon have been fond; Mc Donough [5] shows that every non-abelian free group of at most countable rank admits a faithful permutation representation which is highly transitive. The aim of the present note is to point out that for very group G acting on a set Q there is a permutation group (G*, S2*) with G c 0* and Q c Q* which is highly transitive and such that max (1G1, 1 21) ---- max (10* 17 1Q*1). In fact, we shall see that we may endow (G*, Q*) with the following property of homogeneity: let (A, DA ), (B, S2B) be finitely generated subactions of (0*, Q*), i.e. finitely generated subgroups of 0* together with finitely many of their orbits in D*, then, for every isomorphism yo: (A, DA) -->- (B, DB), there exists an element g g(q9) in 0 inducing this isomorphism. It is a remarkable though in principle well-known fact that a countable homogeneous permutation group is determined up to isomorphism by the isomorphism classes of its finitely generated subactions. If (G, D) is any action of the group G on the set Q there is no problem in finding a set Q* containing Q and a group G* containing 0 and acting faithfully and highly transitively on D* such that (G, D) is a subaction of (0*, Q*). To obtain Q* adjoin a copy of G to kf-2 on Indirizzo dell'a.: Mathematisches Institut der Universitat - Albertstrasse 23b - D-7800 Freiburg i. Br. (Germania Occidentale).

3 296 Otto H. Kegel which G acts regulary from the right ; now (G, S2*) is a permutation group. Let A denote the group of all even finite permutations of 9*, then G*= GA acts as a highly transitive group of permutations on 9*. If (0,,52) is a permutation croup and H is a group containing G then it is possible to find a set 9* containing Q on which H acts faithfully so that (0, 9) is a subaction of (H, Q*). For this let T be a right transversal from G to H with 1 e T; for every t e T choose a copy.12t of Q and consider the disjoint union 9* = U [2t; identify Q with DI. tet If for t e:t and h e H one has th = gt' for some g E G, then the action of H on 9* is described by ( co* = ( co t)h = co sh = woe = (airot' if w* c Qt. Clearly (G, 9) is a subaction of the permutation group (H, 9*) obtained by inducing up. Assuming for the moment that every (abstract) group may be embedded into some (abstractly) homogeneous group without rasising the (infinite) cardinality, we now want to describe a way how to find for every infinite permutation group (G, 9) a highly transitive and homogeneous permutation group (G*, 9*) having (G, 9) as a subaction and of the same cardinality Realise (G, 9) as a subaction (of the same cardinality) of the highly transitive permutation group (G i, 9,). The group 01 may be embedded into a homogeneous group 02 of the same cardinality; inducing the action of 0, on Di up to G, one obtains a set 92 on which 02 acts faithfully and such that (0i, 91) is a subaction of (G2, 92). Any isomorphism g) between two finitely generated subactions (A, OA and (B, 9B) of (0 92) consiats of two parts (2) = = (921, 992) where 99, is the isomorphism of A onto B and cr, is a bijection of DA onto DB so that for 0) E 9, one has (coa) co = ( 0 ) 992 (a)992)a9'16 Since the group G2 is homogeneous, one may assume that the isomorphism Ti is already the identity map. In the symmetric group on 92 there is a element centralising A and inducing the map Form the group G, by adjoining elements of the symmetric group on 92 to G2 one for every isomorphism between finitely generated subactions of (G 92). Put Thus, we have described the first three steps of a construction of larger and larger permutation groups, all of the same cardinality. Iterating this procedure, we obtain permutation groups (G, 9 ) for all natural numbers n, all of the same cardinality and such that for m < n, the permutation group (G m, Sim) is a subaction of (G., Sin);

4 Examples of highly transitive permutation groups 297 if n is of the form n 3k ± 1 then (G., D.) is highly transitive, and for every isomorphism between finitely generated subactions of (G., D.), for n of the form n 3k ± 2, there is a element of (In+, inducing this isomorphism. Thus, putting 0* = U Ga and,q* U,Q., one obtains neri nag a permutation group (0*,,Q*) with all the desired properties. The embedding of every infinite abstract group into some homogeneous group probably was known to P. Hall, it is an immediate consequence of B. A. F. Wehrfritz's construction in [3], chapter VI, of the constricted symmetric group Cs(G) of a group G (compare also [6], 9.4). The group Cs(G) is defined as follows: Cs(G) = fa e symmetric group on the set G; for some finitely generated subgroup F F(a) of G one has (gf)x gf for all g E G}. The group Cs(G) contains all finite permutations on 0 as well as the multiplications from the right by elements of G; it is locally finite if and only if G is locally finite. Let Ge be the group of all right multiplications with elements of G, then for every isomorphism between finitely generated subgroups of Ge(z-2 0) there is an element a e Cs(0) inducing this isomorphism. Thus, by an obvious inductive procedure, one embeds the infinite group 0 into a homogeneous group H of the same cardinality. Going through the preceding argument one sees that if in (0, Q) the group 0 is locally finite all the steps may be performed so that Gn again is locally finite for every n, thus, G* may be obtained locally finite. Summing up, one has THEOREM 1 Let (0, 12) be an infinite (locally finite) permutation group. Then there is a (locally finite) highly transitive and homogeneous permutation group (0*, S2*) of the same cardinality which has (G,,Q) as a subaction. One invariant of a permutation group (G, Q) is its skeleton sk(g, f2), i.e. the set of isomorphism types of finitely generated subactions of (G, Q). The skeleton determines a countable homogeneous permutation group up to isomorphism. This is proved by the standard back-andforth argument of Cantor. THEOREM 2 Let (0, Q) and (H, L') be two countable homogeneous permutation groups. (0, Si) is isomorphic to (H, E) if and only if sk(g, Q) = sk(h, L').

5 298 Otto H. Kegel PROOF. Let (G 00 and (H Ei) be ascending sequences of finitely generated subactions of (0, 0) and (H, E), respectively, with G =U Gi, S2 = Li S2 6 and H -= II H E LiE. Since the skeletons ien ien ien LEN are the same, there is an isomorphic embedding g),. of (G 01) into (H, 2"); choose (H,,, Er ) minimal among the (Hi, Ei ) to contain (61, Si1). as well as (Hi., El). Choose an isomorphic embedding y i of (H1,, Ev ) into (G, S2) and (02., DO minimal among the (Gi, Qi) to contain (H,,, E1, ) 1 as well as (02, 02). By the homogeneity of (G, S2) the map tp1 can be so chosen that the composed map gp lipi is the identity map on (G I21).... Now assume the maps 92: (G, (H, E) and V.: E) -->- (G, S2) so chosen that for i < n' the map 99, +, extends (pi and y i+, extends y i and that the composite maps g) ip. and are the identity maps on (G,, Ow) and (H(n-I) 7 1(n-1)07 respectively. Choose (G( + ) 0.+1),) minimal among the (G D i) to contain (H,, as well as (G.+1, Sin.+1) and a map 99. 1_1 : (0(n + I)' 7 DOH-1Y ) (HI E) extending 99n such that y4).+1. is the identity map on (H,, En, ). Choose (H(n+1), Lfn-1-1),) minimal among the (Hi, Ei ) to contain (G( n+iy Da99 (±1)')n+I as well as (H.4. 1., En+i). Further choose y,, + : (H( n+1) 7 E(n+1)) (07 9 ) as an embedding such that go wn+1 is the identity map on... Denote by g) the limit of the 99n and put y yn. Then g) is an isomorphic embedding of (0, 0) into (H, E) and y is an isomorphic embedding of (H, E) into (G, 9). But, as by constrution the composite maps goy and yg) are the identity on (G, 9) and (H, E), respectively, the maps g) and y must be onto. They are the required isomorphisms. For a E 9 denote by Go, the stabliser of a in the group G acting on 9. With this notation, one obtains the COROLLARY Let (G, 9) be a countable homogeneous permutation group such that every finitely generated subgroup of G leaves infinitely many points of 9 fixed, then (G, 9),S2\cc); if 0 acts transitively on 9 it acts highly transitively. PROOF. The last statement follows from the first by obvious induction. Let (F, 9,) be any finitely generated subaction of (G, 9) then, by assumption, there is an element g E 0 making Fg a subgroup of Go, and thus (F, 9,) isomorphic to a subaction of (0, S2\oc). Theorem 2 now yields the isomorphism. REMARK. Given a countable homogeneous permutation group (G, 9) and an uncountable cardinal x, one may ask the question: is

6 Examples of highly transitive permutation groups 299 there a homogeneous permutation group (H, E) of cardinality x with sk(g,,s2) sk(h, E), Sometimes this question may be answered affermatively using the technique of Shelah and Ziegler [7]. The permutation group (G,,Q) is called existentially closed in the class of all (locally finite) permutation groups if (it is locally finite and) every finite set of equations and inequalities (between elements) in the (first order) language of permutation groups which can be satisfied in some (locally finite) permutation group (H, E) containing (G, Q) as a subaction, can already be satisfied in (G, D). It is easy to see (cf. Hirschfeld and Wheeler [2]) that every infinite (locally finite) permutation group (G, Si) is contained in some existentially closed (locally finite) permutation group (G*,.C2*) of the same cardinality We have same cardinality. We have seen in the proof of theorem 1 that one may enlarge the group G without paying too much attention to Q; thus the group G* will be existentially closed in the class of all (locally finite) groups. On the other hand, n-fold transitivity can be expressed in the (first order) language of permutation groups, and by theorem 1 every (locally finite) permutation group may be embedded into some homogeneous and higly transitive (locally finite) permutation group. Thus, every permutation group (G,,S2) which is existentially closed in the class of all (locally fiite) permutation groups is homogeneous and highly transitive. In this way, we have found for some existentially closed groups in the class of all (locally finite) groups a homogeneous and highly transitive permutation representation. Does every such group admit such a representation? As there is only one countable locally finite group U which is existentially closed in the class of all locally finite groups, namely P Hall's universal locally finite group (see P. Hall [1]; Kegel and Wehrfritz [3], Chapter VI; Mac Intyre and Shelah [4]), the preceding discussion shows that U has a highly transitive permutation representation. Are there many such highly transitive permutation representations of U, or is there only one? REFERENCES [1] P. HALL, Some constructions for locally finite groups, J. London Math. Soc., 34 (1959), pp [2] J. HIRSCHFELD and W. H. WHEELER, Forcing, Arithmetic, and Division Rings, Lecture Notes in Mathematics, vol. 454, 1975 Springer, Berlin.

7 300 Otto H. Kegel [3] 0. H. SEGEL and B. A. F. WEHRFRITZ, Locally finite groups, North-Holland, 1973 Amsterdam. [4] A. MAC INTYRE and S. SHELAH, Universal locally finite groups, J. Algebra, 43 (1976), pp [5] T. P. Mc DONOTJGH, A permutation representation of a free group, Quart. J. Math. Oxford (2), 28 (1977), pp [6] D. S. PASSMAN, The algebraic structure of group rings, John Wiley & Sons, New York [7] S. SHELAH and. M. ZIEGLER : Algebraically closed groups of large cardinality, The J. Symbolic Logic, 44 (1979), pp Manoscritto pervenuto in redazione il 15 luglio 1980.