Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length lambda(G) as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for lambda(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that lambda(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that lambda(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length lambda (p) (G) is introduced, and it is proved that lambda (p) (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-p-soluble length lambda (p) (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author's paper Multilinear commutators in residually finite groups, Israel Journal of Mathematics 189 (2012), 207-224.
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机译：每个有限群G都有一个正态序列，其每个因子都是可溶的或是非阿贝尔简单群的直接乘积。我们将不溶长度lambda（G）定义为此类最短序列中不溶因子的数量。 lambda（G）的上限出现在有限，残差有限和有限群上各种问题的研究中。我们证明，lambda（G）受G的可溶子组的最大2个长度限制，而lambda（G）受可溶子组的最大Fitting高度限制。对于奇数质数p，引入了非p可溶长度的lambda（p）（G），并且证明了lambda（p）（G）不会超过p可溶子组的最大p长度。我们推测，对于给定的素数p和给定的适当基团变体V，Sylow p-亚基属于V的有限基团G的非p-可溶长度λ（p）（G）是有界的。在本文中，我们证明了对于任何可溶品种和有限指数品种的乘积的猜想。作为所得结果的应用，在第二作者论文“剩余线性组中的多线性换向器”的主要结果的证明中，纠正了一个错误，Israel Journal of Mathematics 189（2012），207-224。
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