# Groups with Prescribed Quotient Groups and Associated Module by L. Kurdachenko;J. Otal;I. Subbotin

By L. Kurdachenko;J. Otal;I. Subbotin

...this booklet offers well-established effects in addition to more moderen, modern achieve-ments during this region from the typical indispensable standpoint

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16. Finally we will prove (4). Assume that some quasi - socle of G does not coincide with the socle of G. This means that G includes a normal infinite cyclic subgroup C. Put H = CG(C); so that \G : H\ < 2. 6 yields that A includes a simple DH - submodule B such that A = B © Bx for some x e G\H. If we assume that C n C//CB) * < 1 >, c n cH(Bx) = cn (x~l)cH(B)x *< I > . It follows that C n CH{B) = < 1 >. 21. This final contradiction shows that our assumption does not occur and proves (4). A group G is called hyperfinite, if it has an ascending series of normal subgroups with all factors finite.

Proof Let M b e a minimal G - invariant subgroup of P. Since P < FC(G), M is finite, and therefore, H = CQ{M) has finite index in G. 6 there exists a finite subset X £ G and a simple FH - submodule B such that A = @xeXBx. Since M i s normal in G and CQ{A) = < 1 >, CA(M) is a proper FG - submodule of A, so that CA(M) = < 0 >. 2 implies that charF =£ p. Let K be a finite G - invariant subgroup of P,0 * a e A,A i = aFK. Since K is finite, dimpA \ is also finite. Therefore A \ includes the simple FK - submodule B.

Cy = S such that Ca+i = Ca x Sa+i where Sa+i = Sx„, for some Aa+i e A, a < y. We will apply induction on a. Let a = 1. 2ofthebook[DH]. Let a > 1. Suppose that the Lemma is proved for all ordinal j3 < a. Let a = P + 1 for some p. Then there is a simple FCp- module Vsuch that CcpiV) = < 1 >. Also there is a simple FSp+i - module B such that Cs^ (B) = < 1 >. Let U = V(BF B. Then U = ®xeX Vx for some subset X c Sp+\. We can consider U as an F(Cp xSp+\) - module by Corollary B. 12 of [DH]. Let A be a composition F(Cp+\) - factor of U.