Kovács, L. G.

Neumann, Bernhard Hermann

English

Kovács L. G.

Neumann Bernhard Hermann

University of Szeged

On the existence of Baur-soluble groups of arbitrary height By L. G. KOVACS and B. H. N E U M A N N in Canberra (Australia) A hierarchy of generalized soluble groups has been defined by HERIBERT BAUR. in [1]: abelian groups are to be termed Baur-soluble of height 0; for an ordinal /7 other than 0, a group is to be called Baur-soluble of height h if it is not Baur-soluble of height less than h and has an invariant system (in the sense of KUROSH [3]) whose factors are all Baur-souble of height less than h. Thus Baur-soluble groups of height. 1 are precisely the non-abelian 57-groups. It is easy to show, by induction on h, that . , (1) subgroups and factor groups of Baur-soluble groups of height h are Baur-soluble of height at most It. , A further simple observation is that (2) every restricted direct product of Baur-soluble groups is Baur-soluble, and. its height is the least upper bound of the heights of the factors. The purpose of this note is to demontstrate the existence of Baur-soluble groups-of height /7, for every ordinal h. In fact, the following result will be proved, by induc-tion on h: T h e o r e m . There exist perfect Baur-soluble groups of height h,fi>r every ordinal h. (A group is called perfect if it is its own commutator subgroup.) P roof . The initial step: for h = 0, the unit group is the only example. Obser-vation (2) arid the fact that restricted direct products of perfect groups are perfect take care of the limit step in the induction. I t r e m a i n s to construct, from any given perfect Baur-soluble group G of height h, a perfect Baur-suloble group of height h-f 1. T h e c o n s t r u c t i o n is b a s e d o n w r e a t h powers, which were introduced and studied by HALL in [2]; his notation is adopted throughout. Important use is made of a torsion free, characteristically simple group A constructed by MCLAIN in [5]. This group has the additional property that it is the pro.duct of its abelian normal subgroups. Thus, firstly, A is a perfect ST-group, •and secondly, A, is locally nilpotent. Therefore, firstly, A is a perfect Baur-soluble group of height. 1, and so there is no need to consider the case h = 0. Secondly, as a torsion free locally nilpotent group, A can be (fully) ordered (cf. MAL'CEV [4], or Corollary 6. 2 of B. H . NEUMANN [6] ) . ,Le t A be the ordered set of elements of \A, and let W be the wreath power Wr HA. For each element k of A, HALL defines (in section 2. 4 of [2]) a subgroup D; of IF and shows that (3) each Dx is a direct product of isomorphic copies of H\ (4) the Dk generate (V; and 144 L. C . Kovács and B. H. Neumann: Baür-soluble groups of arbitrary height (5) D-, normalizes Dtí whenever p. 6 A, /JSZ Define Dr, for.each lower section r of A, to be the subgroup of I F generated by the Dk with X£T. Contrary to HALL'S • convention, allow the empty set 0 as a lower section, of A, and put D0 equal the unit subgroup. It follows from (4) and (5) that the Dr are all normal subgroups of W. As unions of ascending chains of lower sections, as well as intersections of any number of lower sections, are again lower sections, it follows that the Dr form an invariant system of W. Let DA/Dr be a factor of this system; then no member of the system can lie properly between Dr and DA , and so no lower section properly contained in A can properly contain f . From this it is easy to deduce that A contains just one element X which is not contained in F. Hence DA—DrDx, so that DJDr is isomorphic to a factor group of Dk. Now (1), (2) and (3) give that each factor of this invariant system of W is Baur-soluble of height at most h, and therefore W is Baur-soluble of height at most /7 + 1. On the other hand, as W is generated by the perfect Baur-soluble groups D, of height /?, W is perfect, and its height is at least h. If the height of IK is /7+ 1, the construction is complete. Assume that W is of height /7. .Let the elements of A act on A as right translations; then A becomes a transitive and hence irreducible group of order-preserving permutations of A. Take G to be the natural split extension of W by A. (In [2], HALL takes A to be the group of all order-preserving permutations of A, but the omission of "all" does not effect his subsequent results.) Since G/W^A and A is Baur-soluble of height 1, G. is Baur-soluble of height at most /7 + 1; also, as If and A are perfect, so is G. It follows from Theorem D of HALL [2] that W is a minimal normal.subgroup of G; therefore every invariant system of G has a factor which contains an isomorphic copy of W and therefore cannot be Baur-soluble of height less than H. Consequently, the height -of G cannot be less than h + 1. This completes the construction in case W is of height h. References [1] HERIBERT BAUR , Ein Kommutativitätskriterium für unendliche auflösbare Gruppen, Arch. Math., 11 (1960), 176-182. [2] P. HAUL, Wreath powers and characteristically simple groups, Proc. Cambridge Philos. Soc., 58 (1962), 170-184. [3] A. G. KUROSH, Theory of groups, vol. 2 (New York, 1956). [4] A. I. MAL'CEV, On the completion of group order, Trudy Mat. Inst. Steklov, 38 (1951), 173 — 175 (Russian). [5] D . H . MCLAIN, A characteristically-simple group, Proc. Cambridge Philos. Soc., 5 0 ( 1 9 5 4 ) , 641 — 642. [6] B. H. NEUMANN, An embedding theorem for algebraic systems, Proc. London Math. Soc., (3) 4 (1954), 138-153. THE AUSTRALIAN NATIONAL UNIVERSITY (Received April 10, 1964) 

On the existence of Baur-soluble groups of arbitrary height

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On the existence of Baur-soluble groups of arbitrary height

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