filters and partitions of an infinite set Q. The structure of subgroups, in particular maximal subgroups, of finite symmetric groups was investigated by M. O'Nan ...
MAXIMAL SUBGROUPS OF INFINITE SYMMETRIC GROUPS MARCUS BRAZIL, JACINTA COVINGTON, TIM PENTTILA, CHERYL E. PRAEGER, and ALAN R. WOODS [Received 2 July 1991—Revised 2 November 1992]
1. Introduction By a maximal subgroup, we mean a maximal proper subgroup of the symmetric group S = Sym(Q) of all permutations of a set Q. Maximal subgroups of finite symmetric groups have been studied intensively; however our work was largely motivated by a recent paper [7] of Dugald Macpherson and Peter Neumann in which they considered maximal subgroups when Q is infinite. (Some of the results there were anticipated by Ball [2] and Richman [9] in early papers.) Taking [7] as our starting point, we have extracted what we believe to be the fundamental concepts. This has led to several new constructions, and in some cases, classification, of maximal subgroups which lie above the stabilizers of subsets, filters and partitions of an infinite set Q. The structure of subgroups, in particular maximal subgroups, of finite symmetric groups was investigated by M. O'Nan and L. L. Scott, see [10], and later by M. Aschbacher and Scott [1]. The approach used by them was extended and, in some sense completed, by Liebeck, Praeger and Saxl [6], to show that maximal subgroups of finite symmetric groups are well understood in the sense that there are several types of maximal subgroup, and their structure is completely determined modulo the classification of finite simple groups. More precisely, given a maximal subgroup of a finite symmetric group, either it is the (setwise) stabilizer of a subset, partition, product decomposition, or affine structure; or it is a group in diagonal action; or it is almost simple, that is, lies between a non-abelian simple group and its automorphism group. The original aim was to find new classes of maximal subgroups of infinite symmetric groups which were in some sense analogues of classes of maximal subgroups of finite symmetric groups. Ball [2] showed that the setwise stabilizer 5 { r ) of a finite subset F is maximal in 5. But Neumann and Macpherson proved that any maximal subgroup which is not the stabilizer of a finite set is highly transitive, and hence cannot stabilize a partition or preserve a geometry. Thus at first sight it seems that almost none of the classes of maximal subgroups of finite symmetric groups have analogues in the infinite case. However, one can still ask for maximal subgroups containing the stabilizer of a set, partition, etc. Typically these must contain all permutations which in some sense almost stabilize the structure. For instance, Richman [9] showed that the almost stabilizer of a partition of Q into finitely many parts of equal cardinality is a maximal subgroup, and we investigate further maximal subgroups which are, or lie above, the almost stabilizers of subsets, filters and partitions. The key to understanding these is the study of groups which contain the pointwise stabilizer 5 (A) of some set A is called proper if it is a proper subset of &(Q), that is, if Q i 2,. Ideals and filters are dual in the sense that ^ is a filter if and only if ^ = {A: Q - A e f } is a proper ideal. Note the asymmetry in the convention that we allow improper ideals but not improper filters. A collection 38 of subsets of Q is a filter base for 9 if & = {2 c Q: A c 2 for some A e 38}. Then we say that SF is the filter generated by 58. A collection 38 is a filter base if and only if the intersection of any two sets in 38 contains some element of 38. Consequently, any collection 38 which is closed under intersections is a filter base. A filter generated by a single set A is called a principal filter; it has the form ^ = { I c Q : A c l } . Recall that the depth A(^) of a filter 9 is given by A(^) = min{|A|: A e 9). Note that every filter of finite depth is principal. A uniform filter is a filter of depth K, that is, one in which every member has cardinality K. The filters we are
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interested in in this section correspond to filters over the quotient of the Boolean algebra of subsets of Q by the equivalence relation =A. For this reason, we refer to filters in which A = K Q for every element A in this filter as trivial filters. These are just filters which contain no moieties. Obviously if $F is a non-trivial filter and ^ is a filter with f c = {jce5: Tx e f o r e 9 for all T c Q}. The stabilizer of a filter f is a subgroup of S and is a proper subgroup unless SF = {A: |Ac| < A} for some A such that Ko =s A =s K. In the case where A = No, this is the Frechet filter of all cofinite sets. These filters are all trivial, so the stabilizer of any non-trivial filter is a proper subgroup of 5. The stabilizer of an ideal $> is defined in a similar manner. If 3> is the dual ideal to a filter $F then 5{gf} = S w . Notice that a permutation g belongs to 5{gt:} if and only if both Ts e 2F and P " ' e f for all F e f . However, to show that a group G is a subgroup of S{3J:} it suffices to show that T8 e SF for any g e G. Observe also that if we have a filter base 58 then it suffices to consider only those sets r e 58. Simon Thomas [3] has shown that it is consistent with ZFC that there are subgroups of infinite symmetric groups which are not contained in any maximal subgroup. However, Macpherson and Neumann have proved: PROPOSITION 3.3 (Macpherson and Neumann [7]). Let G be a proper subgroup of S. If there is a moiety HofQ such that G{X} induces the full symmetric group on 2 then there is a maximal subgroup H of S such that G =£ H.
One way to satisfy the hypothesis of the proposition is to require the stronger condition that 5 ( A ) ^ G for some A with |AC| = K. It can easily be seen that this condition is actually stronger—moreover, Covington and Mekler [4] have recently shown that this condition is stronger even when G is assumed to be maximal. The condition is satisfied, for example, if G is the stabilizer of a finite partition. (To see this, take A to be the union of all but one of the parts.) We have already seen in Proposition 3.1 that these subgroups are maximal. Richman [9] was perhaps the first to see the significance of Stabilizers Of filters in the study of maximal subgroups, but he was only concerned with stabilizers of collections of ultrafilters. Then Semmes [11] proved (cf. Theorem 3.5 below) that if H is maximal subgroup such that 5 (A) ^H for some A with |A| < K, then there is a filter 2F such that H = S{3f). We will continually exploit this representation, which we show to be unique. However, we will also be interested in the case where |A| = K. Then it is not necessarily true that H is the stabilizer of a filter. However, as we show below, if it is not, then H is the almost-stabilizer of a finite partition of Q. In order to prove this we introduce the notion of a quasifilter. A collection 2. of subsets of Q is called a quasifilter on Q if it has the following
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properties: (a) 0^andQeS; (b) if T e 2 and T . Thus it remains to show that 5 { 2 } =£S. Take A e S with |Ac| = *:and choose F c A c such that |r| = |A| and A U I V Q . Then r $ 2, so there is no g e S{&) such that A8 = T. Thus G ^ 5 { 2 } < 5. Finally, if G is maximal, we must have G = 5 {2} . In the special case where |A| < K, we obtain: THEOREM 3.5 (Macpherson and Neumann [7]). / / G is a proper subgroup of S with 5 (A) ^ G for some set A such that \ A| < K, then G^S{^) is the non-trivial filter generated by thefilterbase
m = {A:
\A\
Proof. Let 2, and 2 2 be moieties in 9. If |2, - 2 2 | = | 2 2 - 2 i | , then, by Lemma 3.6, there exists g e S^n^y^SiSF) interchanging 2 , - 2 2 and 2 2 —2^ Hence we can assume \2.x — 2 2 | < |2 2 - 2,|, and so |2, fl 2 2 | = |2,| = K. Since 9 is not almost principal, there is some A e ^ which is a moiety of 2 , f l 2 2 . By Lemma 3.6, Sym(A c )^5 {gf }, and hence there exist g, h e5 (A) ^5{y} such that 2f = 2 2 and 2j = 2,.
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5. Almost stabilizers of partitions We now return our attention to maximal subgroups containing the stabilizer of a partition. Let 9 = {Qt: i el} be a partition of Q into /i = |/| parts each of size |Q,| = A. Then *: = A/*. When Q is finite, S{g>) is a maximal subgroup for any p>l, A> 1. The case where Q is infinite is much more complicated. We show in this section that in all cases a suitably defined almost stabilizer is maximal. We have already seen (Proposition 3.1) that if 9 is a finite partition, that is, if /i is finite, then the almost stabilizer of 9* is maximal. Macpherson and Neumann [7, Observation 6.3] constructed a maximal subgroup containing S{9) in the special case where n = |/| is a regular infinite cardinal such that /i < cf(*-) (and therefore A = K). For each g e S, let We can think of this as the number of parts over which g spreads QJ} excluding parts which receive almost no elements. They defined a maximal subgroup H by H = {g: ty(g) < \i and ^(g" 1 ) < \x for all y e / } . Our first result applies to arbitrary partitions with an infinite number of infinite parts of the same size, and includes these subgroups of Macpherson and Neumann as a special case. Recall that if 9* = {Q,: / e /} is a partition of Q into \i parts, each of cardinality A, where A, ju are infinite, then AStab(0>) = {geS: (Vi e I)(Bj, k e /)(Q? = A Q, A Qf ' = A Q*)}. THEOREM 5.1. Let 9 = {Q,: / e /} be a partition of a set Q of cardinality K into fj, = \I\ parts, each of cardinality |Q,| = A, where both A and fj. are infinite. For any S c Q , let
and define Then 2F is a filter and S{3F} is a maximal subgroup of S = Sym(Q) containing AStab(^). A set 2 lies in 2F if it 'almost contains almost all the parts of 9*\ First we prove that SF is a filter and establish a few of its properties. LEMMA
5.2. The filter 9 is a non-trivial uniform filter on Q with
Proof It follows from the definition of & that Q e ^ and every superset of a set in 9 also lies in 9, and it is easy to see that 0 £ 9. Now choose 2, T e SF. Now, N(l H T) = N(l) U N(T) and both N(X) and N(T) have cardinality less than ju, whence 2 fl T e 9. Thus ^ is a filter. Moreover, each 2 e ^ contains A points of each of /i parts of 9, and hence |2| = jtfA = *:. Therefore ^ is uniform. To show that 2F is non-trivial we construct choose a point c} e Qy. Let a moiety in $*. Choose iel, and for all jel-{i} 2 = Q - (Q, U r), where T = {c;: j ± /}. Then N(2) = {/}, so 2 e 9, and |2 C | = max{A, JU} = K SO 2 is a moiety in the filter.
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If g e AStab(^) then |N(2)| = |N(2*)|, so 2 e ^ if and only if 2* e &. Thus g e 5{SF}, and so AStab(^) ^ 5{Sf>. Proof of Theorem 5.1. Suppose that S{g;}x = {Q?: / € / } . Clearly S{9*}^H. moiety L of J and for each j € L choose a point bj e Qy - {ay}. Then H contains a permutation z which fixes Qy pointwise for j eJ - L, and fixes Qy setwise and interchanges ay and bj for each j e L. Now (K{r))z is full for Tz. Also, z fixes Qy setwise and so does K{r^; hence (K{r))z fixes Qy setwise, and fixes Qy — F z pointwise. Thus if we let AT, be the subgroup of 5 ' induced on Qy by Kz, then Now, since TC\rz = {dj\ jeJ — L] has cardinality K, we can deduce by applying Proposition 2.1 to Qy that (Ko, /C,) = S y m ( r u P ) . Using a similar argument several times with different choices of the points 6y and the moiety L of / , we obtain the full symmetric group 5 ' induced by H on Qy\ Then H is full for Qy\ But since x e H, it follows that H is full for Q y . We can now complete the proof of Theorem 5.9. LEMMA
5.13. Let H be as in Lemma 5.12. Then H = S.
Proof. By Lemma 5.12 there is a moiety / of / such that H is full for Q y . Let L be a moiety of / such that I = JU L and \J D L\ = K. There is an element z eS{&>} such that QL = Qj. Since zeH, H is also full for QL. So by Proposition 2.2, H = S. We leave the following problem open: Let 9* = {Q,: i e 1} be a partition of an infinite set Q into fj. = \I\ parts, each of cardinality |Q,| = A. Characterize all the maximal subgroups of Sym(Q) which contain 5{go>. 6. Filters with a chain as a filter base A fundamental problem which we have not solved is to find a set-theoretic condition on a filter ^ which is necessary and sufficient for its stabilizer 5{3f} to be maximal. Clearly the problem reduces to asking: What are the maximal subgroups lying above the stabilizer S{3?} of a filter &? In the present section we consider this question for filters cF which are 'simple' in that they have a chain as a filter base, that is, there are some linear ordering (/, < ) and filter base 38 = {A,: iel) for 8F such that A, z> Ay whenever i) is a maximal subgroup. (b) If\®\ = K then S{^ < AStab(^) where AStab(^) is the almost stabilizer of the partition $P= {, C}, and AStab(^) is the only maximal subgroup lying above S{§).
In the case where a filter $F is not almost principal and has a chain as a filter base, the question of what maximal subgroups lie above S{3f) is settled by: THEOREM 6.2. Let \Q\ = K and let & be a filter on Q which is not almost principal and has a chain as a filter base. Then the stabilizer 5{3F+} of the superclosure cF+ of cF is a maximal subgroup lying above 5 { ^ } . Also the following hold. (a) / / SF = 0 , there are no other maximal subgroups above S^). (b) / / p=t0 then S{Sp} lies below exactly one other maximal subgroup. If |4>SF| < K then this is S ^ = AStab(4>3f). / / |p| = if then it is the almost stabilizer AStab(^) of the partition &={&&,
