关于有限秩的幂零群的自同构
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摘要
设G是有限秩的幂零群,1=ζ_0G<ζ_1G <…<ζ_cG=G是G的上中心列,End(ζ_iG/ζ_(i-1)G)是Abel群ζ_iG/ζ_(i-1)G的自同态环(1≤i≤c),End(ζ_iG/ζ_(i-1)G)可以自然地作成一个Lie环.α_1,α_2,…,α_n是G的n个自同构,把它们在ζ_iG/ζ_(i-1)G上的诱导自同构分别记为α_(1i),α_(2i),…,α_(ni)(1≤i≤c).如果由α_(1i),α_(2i),…,α_(ni)生成的Lie环End(ζ_iG/ζ_(i-1)G)的Lie子环都是完全可解的,那么α_1,α_2,…,α_n生成的AutG的子群具有良好的幂零性质.考虑G的下中心列,可以得到对偶的结果.
LetG be a nilpotent group, and 1=ζ_0 G < ζ_1 G < …<ζ_cG = G be the upper central series of G. End(ζ_iG/ζ_(i-1)G) is an endomorphism ring of an abelian group ζ_iG/ζ_(i-1)G, naturally, End(ζ_iG/ζ_(i-1)G) is a Lie ring. Suppose thatα_1,α_2,…,α_n are automorphisms of G. Denote by α_(1 i),α_(2 i),…,α_(ni) the automorphisms induced by α_1,α_2,…,α_n on ζ_iG/ζ_(i-1)G, then α_(1 i),α_(2 i),…,α_(ni) generate a Lie subring of End(ζ_iG/ζ_(i-1)G). If Lie ring generated by α_(1 i),α_(2 i),…,α_(ni) is completely solvable,then automorphism subgroup generated by α_1,α_2,…,α_n has good nilpotence. In addition, we investigate analogous problems of the lower central series of G and obtain the similar arguments.
LetG be a nilpotent group, and 1=ζ_0 G < ζ_1 G < …<ζ_cG = G be the upper central series of G. End(ζ_iG/ζ_(i-1)G) is an endomorphism ring of an abelian group ζ_iG/ζ_(i-1)G, naturally, End(ζ_iG/ζ_(i-1)G) is a Lie ring. Suppose thatα_1,α_2,…,α_n are automorphisms of G. Denote by α_(1 i),α_(2 i),…,α_(ni) the automorphisms induced by α_1,α_2,…,α_n on ζ_iG/ζ_(i-1)G, then α_(1 i),α_(2 i),…,α_(ni) generate a Lie subring of End(ζ_iG/ζ_(i-1)G). If Lie ring generated by α_(1 i),α_(2 i),…,α_(ni) is completely solvable,then automorphism subgroup generated by α_1,α_2,…,α_n has good nilpotence. In addition, we investigate analogous problems of the lower central series of G and obtain the similar arguments.
引文
[1] Helmut S., Rolf F., Modular Lie Algebras and Their Representations, Marcel Dekker, New York, 1988.
[2] James E. H., Introduction to Lie Algebras and Representation Theory(Third Printing), Springer-Verlag,New York, 1980.
[3] Liu H. G., Zhang J. P., Xu T., On the automorphisms of a nilpotentπ-groups of finite rank(in Chinese),Sci. China Math., 2012, 42:787-802.
[4] Liu H. G., Zhang J. P., Xu T., On the automorphisms of a nilpotent group of finite rank, Acta Math. Sin.,Chin. Ser., 2014, 57(4):625-656.
[5] Robinson D. J. S., A Course in the Theory of Groups(Second Edition), Springer-Verlag, New York, 1996.
[2] James E. H., Introduction to Lie Algebras and Representation Theory(Third Printing), Springer-Verlag,New York, 1980.
[3] Liu H. G., Zhang J. P., Xu T., On the automorphisms of a nilpotentπ-groups of finite rank(in Chinese),Sci. China Math., 2012, 42:787-802.
[4] Liu H. G., Zhang J. P., Xu T., On the automorphisms of a nilpotent group of finite rank, Acta Math. Sin.,Chin. Ser., 2014, 57(4):625-656.
[5] Robinson D. J. S., A Course in the Theory of Groups(Second Edition), Springer-Verlag, New York, 1996.