Cyclical subnormal separation in A-groups

Abstract

Three main results, concerning ,4-groups in respect of cyclical subnormal separation as defined in [4], are presented. It is shown in theorem A that any /1-group that, is generated by elements of prime order and satisfying the cyclical subnormal separation condition is metabelian. The two other main results give necessary and sufficient conditions for r4-groups, that are split extensions of certain abclian p-groups by a metabelian p' group, to satisfy the cyclical subnormal separation condition. There is also a result which shows that .4-groups with elementary abelian Sylow subgroups are cyclically separated as defined in [3]. 1-. Introduction A group G is called a CSn group if for any given cyclic subgroup B '„ group if and only if every element of // of prime order is in the Fitting subgroup of H. Theorem C Suppose that G = W » H is an /1-group where W is a minimal normal subgroup of G and // is a metabelian /)'-subgroup acting faithfully on W. Then G is a CSn group if and only if every elementof H\F(H) acts fixed-point-freely on 14-', where F(!l) is the Kitting subgroup of H. We also come across this result, Lemma (3.3) which concerns CS groups. We find it appropriate to bring it here because the CSpaper introduced in [3] was the motivating factor for the appearance of the CSn groups. There was the general feeling that /l-groups may be C.^-groups. Lemma (3.3) gave one positive result. But as it turns out we have an example of an A group which is even a CS7I group but not a CS-group. Section 2 deals with general results that are of interest to us including an example of a rion metabelian ,4-group in CSn. In fact as it has been pointed out in the concluding remarks ,i slight modification of this example gives an example of an /1-group in CSn which i> not a CS group. MIRAMARE TRIESTE December 1995 'Permanent address: Department of Mathematics, Ahrnadu Bello University, Samaru, Zaria, Kaduna State, Nigeria. 2 Preliminaries 2.1 Theorem A established that any A-group in CSn which is generated by elements of prime order must be metabelian. We begin our discussions with an example to show, first, that there are non metabelian /l-groups in CSn and secondly that the hypothesis, in the theorem, of the generating elements to be of prime order is necessary. We note that metabelian groups had already been shown to belong to the class of CSn groups [(3.5) of [4]]2.2 Example (2.2) Suppose that p, q and r are three distinct primes and (i) is a cyclic group of order r. Let k be the order of q mod r, i.e., r ^ j and r f IJ' — 1 for ( '') = 1Tliis implies that C'JX'O — 1. Clearly, O is an .4 group. To see why it is a f-'.S',, group we first observe tiiat C = L » H where II = M{.r) is a ;/-metabelian group, by [(3.!)) [4]] // G Ci>'n. Nest, any (; eleinent in // is conjugate to some m € M arid by n) denotes the smallest subnormal subgroup of // containing in. Note that. (m) = (m) since A/ is abelian and normal in //. Also, any r element of // is conjugate to an eleiLient in (x). But For 1 / ;/ 6 (•'')-,!> = •>' f° soim; v £ {1,2. . , . , r I [. If i; — r we have seen that f.'r,(x) = I. If » ^ r then y = :(:' and Hence by theorem A of [4] G is a CSn group. The following special case gives an idea of what G looks like. Let r = 2,q = 3 and k — 1 then M = (m) and (,r) are cyclic of orders !J and 1 respectively. Also M(.r> = {m,.r|m = J= l,m* = m'} H