A Transformation Group and Its Subgroups (trilogy of theory of dimensionality #2 (full paper)) (original) (raw)
2015, Communications in Applied Geometry
We discuss a transformation as follows. There is a point on the number line, then we move it on an integer m to another one n on the number line. We denote the transformation with mTn. We think of all the elements G = { mTn | m, n in Z } m}Z. Then, we discuss the binary operation lTn = lTmmTn. It denotes the repetition of transformation. G makes a group under the binary operation. This is a modeling of group in [1] on the number line. In this paper, we think of the subgroups. 2010 MathematicsSubjectClassification. 20A10. Corrigendum: In the proof of Proposition 2.1., strictly speaking, the right cosets Hg={2k-1T2n} "where 2n is fixed" and H'g'={2kT2n-1} "where 2n-1 is fixed." Remark. In Proposition 2. 2., {mT0} cup {0Tm} itself can’t be obviously a group. For example, -mT00Tm = -mTm is not in {mT0} cup {0Tm}.
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