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半直积

一个“分裂”扩张 G of groups N and F which contains a subgroup F^_ isomorphic to F with G=F^_N^_ and F^_ intersection N^_={e} (Ito 1987, p. 710). Then the semidirect product of a group G by a group H, denoted H×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]G (or sometimes H:G) with homomorphism T is given by

(g,h)(g^',h^')=(gg^',(h(g^'T))h^'),

其中 g,g^' in G, h,h^' in H, and T in Hom(G,Aut(H)) (Suzuki 1982, p. 67; Scott 1987, p. 213). 注意，两个群的半直积不是唯一确定的。

一个群的半直积 G by a group H can also be defined as a group S=GH which is the product of its subgroups G and H, where H is normal in S and G intersection H={1}. If G is also normal in S, then the semidirect product becomes a group direct product (Shmel'kin 1988, p. 247).

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参考文献

Itô, K. (Ed.). 'Extensions." §190.N in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 710, 1987.Kurosh, A. G. The Theory of Groups, 2nd ed., 2 vols. New York: Chelsea, 1960.Scott, W. R. "Semi-Direct Products." §9.2 in Group Theory. New York: Dover, pp. 212-217, 1987.Shmel'kin, A. L. "Semi-Direct Product." In Vol. 8 of Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia" (Managing Ed. M. Hazewinkel). Dordrecht, Netherlands: Reidel, p. 247, 1988.Suzuki, M. Group Theory, Vol. 1. New York: Springer-Verlag, 1982.

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半直积

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Weisstein, Eric W. "Semidirect Product." 来自 —— 资源。 https://mathworld.net.cn/SemidirectProduct.html

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