Formulation of the homological functors of some Bieberbach groups with dihedral point group (IR)
The purpose of this research is to produce the formulas of the homological functors of some Bieberbach groups of dimension ?ve with dihedral point group of order eight. The homological functors consist of the nonabelian tensor square, G-trivial subgroup of the nonabelian tensor square, the central s...
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| Format: | thesis |
| Language: | eng |
| Published: |
2015
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| Subjects: | |
| Online Access: | https://ir.upsi.edu.my/detailsg.php?det=1026 |
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| Summary: | The purpose of this research is to produce the formulas of the homological functors of some Bieberbach groups of dimension ?ve with dihedral point group of order eight. The homological functors consist of the nonabelian tensor square, G-trivial subgroup of the nonabelian tensor square, the central subgroup of the nonabelian tensor square, the nonabelian exterior square and Schur multiplier. The computational method for polycyclic groups is used to determine the formulas of the functors. Selected theorems, lemmas and de?nitions are also used in the computation. The ?findings of this research are the new formulas of the homological functors and also the generalization of one of the functors that is the central subgroup of the nonabelian tensor square. As a conclusion, the nonabelian tensor square and the nonabelian exterior square are found to be not abelian. While G-trivial subgroup of the nonabelian tensor square, the central subgroup of the nonabelian tensor square and Schur multiplier are abelian. The implication of the ?ndings enables the properties of other Bieberbach groups to be explored. In addition, the results can be bene?cial not only to mathematicians in terms of theories and applications but also to chemists and physicists. |
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