A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that $$G=HK$$ G = H K and $$H \cap K=1$$ H ∩ K = 1 . We prove that, for a locally soluble group G , all cyclic subgroups are complemented if and only if it is the semidirect product of groups $$A= {{\,\mathrm{Dr}\,}}_{i \in I} A_i$$ A = Dr i ∈ I A i by $$B={{\,\mathrm{Dr}\,}}_{j \in J} B_j$$ B = Dr j ∈ J B j , where all factors $$A_i$$ A i and $$B_j$$ B j are finite of prime order, and A has a set of maximal subgroups normal in G with trivial intersection. An analysis of the structure of periodic locally soluble groups of infinite rank shows, in particular, that if G is a periodic locally soluble group whose infinite rank subgroups are complemented, then every subgroup of G is complemented.

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具有一些互补子群的群

如果存在 G 的子群 K 使得 $$G=HK$$ G = HK 和 $$H \cap K=1$$ H ∩ K = 1，则称 G 的子群 H 在 G 中是补的. 我们证明，对于局部可解群 G ，所有循环子群都是互补的，当且仅当它是群 $$A= {{\,\mathrm{Dr}\,}}_{i \in I 的半直积} A_i$$ A = Dr i ∈ IA i by $$B={{\,\mathrm{Dr}\,}}_{j \in J} B_j$$ B = Dr j ∈ JB j ，其中所有因素$$A_i$$ A i 和$$B_j$$ B j 是素数阶有限的，并且A 有一组极大的子群在G 中正规并具有平凡的交集。对无穷阶周期局部可解群的结构分析特别表明，如果G是无穷阶子群互补的周期局部可解群，则G的每一个子群都是互补的。