By Liman F.N.
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Extra resources for 2-Groups with normal noncyclic subgroups
14 (). 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 (). 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 ) 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  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.