Structural Analysis of Metallic Glasses with Computational Homology by Akihiko Hirata Kaname Matsue & Mingwei Chen

Author:Akihiko Hirata, Kaname Matsue & Mingwei Chen
Language: eng
Format: epub
Publisher: Springer Japan, Tokyo

We can define the quotient abelian group for any integer in the similar manner. In this case, we identify two integers m and n in if is a multiple of t. Thanks to the fact for any integers , the set with the operation

determines the group structure on . The abelian group is called the quotient group of by . We also write as .

General quotient groups, say G / H for abelian groups, can be constructed in the same manner with careful treatments of group operations. For example, the concept of being homologous in Definition 3.5.3 is realized in the similar manner to the definition of . Once one can understand the identification rule, he or she will easily understand the meaning of the quotient in homology groups (Definition 3.5.5).

In general, for given abelian group G and its abelian subgroup H, elements of quotient group G / H are called residual classes or equivalent classes of G subject to H. For example, an equivalent class of subject to is

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