Engel-type subgroups and length parameters of finite groups
Engel-type subgroups and length parameters of finite groups
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有限群的恩格尔型子群和长度参数
DOI：
10.1017/s1446788719000181
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发表时间：
2015
影响因子：
1
通讯作者：
G. Traustason
中科院分区：
文献类型：
--
作者：
Evgeny Khukhro;P. Shumyatsky;G. Traustason
设g是有限群G的一个元素，对正整数n，设En(G)是x∈G上所有交换子[…[x，g]，g]，…，g生成的子群，其中g重复n次.根据Baer定理，如果En(G)=1，则g属于拟合子群F(G)。我们用En(G)的某些长度参数推广了这一定理。对于可解G，我们证明了如果对某个n，EN(G)的拟合高度等于k，则g属于第(k+1)个拟合子群Fk+1(G)。对于不可溶G，其结果是以不可溶长度和广义拟合高度表示的。有限群H的广义拟合高h*(H)是使fh*(H)=H的最小数，其中F0*(H)=1，Fi+1(H)*是广义拟合子群F*(H/F*i(H))的逆像.设m是|g|计数重数的素因数个数。证明了对于某个n，如果En(G)的广义拟合高度等于k，则g属于F*f(k，m)(G)，其中f(k，m)仅依赖于k和m.有限群H的不可解长λ(H)被定义为正规序列中每个因子要么是可解的，要么是非交换单群的直积的最小不可解数.证明了如果λ(En(G))=k，则g属于不可解长关于k和m有界的正规子群.我们还提出了与m无关的更强结果的猜想，并证明了这些猜想归结为有限单群直积的自同构问题.
Let g be an element of a finite group G. For a positive integer n, let En(g) be the subgroup generated by all commutators [...[[x, g], g],..., g] over x ∈ G, where g is repeated n times. By Baer’s theorem, if En(g) = 1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of En(g). For soluble G we prove that if, for some n, the Fitting height of En(g) is equal to k, then g belongs to the (k+1)th Fitting subgroup Fk+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that Fh* (H) = H, where F0* (H) = 1, and Fi+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of En(g) is equal to k, then g belongs to F*f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(En(g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.