“semi-direct product”的概念、定义、习题解题方法及翻译-数学辞典

单词

semi-direct product

释义

semi-direct product

A group G is an internal semi-direct product of a normal subgroup N and subgroup H if G = NH and N ∩ H = {e}. This is written G = N ⋊ H. In this case every g ε G can be uniquely written g = nh, where n ε N and h ε H. Multiplication in the group is given by

Such an example is the dihedral group D2n, where N is the subgroup of rotations and H is a subgroup generated by a reflection.

For h ε H, the map φh: n ↦hnh-1 is an automorphism of N and φ: h ↦ φh is a homomorphism from H to Aut(N). More generally, given two groups N and H and a homomorphism φ: H→Aut(N), the external semi-direct product N⋊φH can be formed by multiplying elements of the Cartesian product according to rule

The semi-direct product is a means of creating a larger group from two groups other than the direct product. If the map φ is the constant map to the identity map of N, then the semi-direct product agrees with the direct product.

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