By Liman F.N.

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**Extra resources for 2-Groups with normal noncyclic subgroups**

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14 ([55]). Let G be a locally generalized radical group and T be the maximal normal periodic subgroup of G. If G has a ﬁnite 0-rank r, then G/T has ﬁnite special rank. Moreover, there is a function f5 : N −→ N such that r(G) ≤ f5 (r). Indeed we can put f5 (r) = r + f2 (r). 15 ([55]). Let G be a locally (soluble-by-ﬁnite) group. If G has a ﬁnite 0rank r, then G has normal subgroups T ≤ L ≤ K ≤ S ≤ G such that T is locally ﬁnite, L/T is nilpotent and torsion-free, K/L is abelian torsion-free and ﬁnitely generated, G/K is ﬁnite and S/K is soluble.

It follows that the family of all non-zero RH-submodules of A has a minimal element B. Then B is clearly simple. Let X be a transversal to H in G so that |X| = n. 4, A = g∈X Bg. 3. 6. Let R be a ring, G a group, and H a normal subgroup of G of ﬁnite index. If A is a semisimple RG-module, then A is a semisimple RH-module. This extends the celebrated theorem of Cliﬀord (H. Cliﬀord [45]) to inﬁnite groups. I. 5. Chapter 5. 7. Let F be a ﬁeld, G an abelian group, and H a periodic subgroup of G. If A is a simple F G-module, then A is a semisimple F H-module.

Since fβ is surjective, Aβ+1 ≤ B. Equality is clear. It now follows that if T is an arbitrary submodule of A and β is the largest ordinal such that Aβ ≤ T , then every non-zero element of T has height less than β; that is, T ≤ Aβ . Hence T = Aβ , and the proof is complete. B. Hartley [105] has also constructed examples of uncountable artinian modules over group rings. For the reader’s convenience, we mention here some of his important results without proof. Let p = q be primes, and let Q be a Pr¨ ufer q-group.