By Ji L.
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Additional info for 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four
But then H satisfies rnin by hypothesis, and thus H i s finite. Consequently, H = Hk for some k, and so if k c j < i one has Zi _c Hin ZiG Z j . Put Z = n i , k z i . Then since each of the subgroups Zi is finite and nontrivial, we have Z # 1, and the subgroup Z is contained in the centre of U i z k G i = G. This is the desired contradiction. 5 one cannot weaken the assumptions by replacing ,,ascendant” by ,,subnormal”, for in P. Hall a construction is given of a countably infinite, locally finite p-group, for any prime p, in which (1) is the only abelian subnormal subgroup.
If c E C, then there exists an id = j(id-'j)d = id2 = jc. Thus all the elements of D2,\C are conjugate in D z m . The subgroup C being locally cyclic, contains only one involution, z say, and z is central in D Z msince z' = z-l = z. It is defined by Qzn= (x, y ; x2 = y 2 = (xy)'" = z, where z 2 = 1). The group Q2 is the ordinary quaternion group of order eight. Clearly, the element z of the (generalized) quaternion group QZn is in the centre of Q2.. One easily shows that the element z is the only involution of the (generalized) quaternion Q,..
Moreover the condition that the group G be locally soluble can be slightly weakened by assuming only that G contains a normal series with abelian factors (that is, G is an SI-group in the sense of Kurog [I], 9 57). 7 Lemma. If the group G has afinite series the factors of which are either finite groups or Prufer groups, then G is a Cernikov group. Proof. l the group G satisfies min. 4 and a simple induction argument G has a finite normal series ... ( l } = H o d K o ~ H l e K l d dHrQKr=G where each factor H i / K i - is abelian and each factor K J H , is finite.
2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four by Ji L.