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2006•Journal of Dezhou UniversityRequires access

The Conditions for Both the Groups Consisting of Two-sided Coset and the Groups Consisting of Products of One-sided Coset

Jingxiao Zhang

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Abstract

The concepts of one-sided coset and two-sided coset were given,some of properties of two-sided coset were discussed,and it was pointed out that two-sided coset b~(-1)Ha~(-1) is just consist of inverse elements of each element existing in two-sided coset aHb,and that aHb≤ Gb~(-1)Ha~(-1)≤G.With the use of the basic properties of one-sided coset and by use of the method of circulating proof,twenty six equivalent propositions on the groups consisting of two-sided coset aHb were proved.Finally,seven necessary and sufficient conditions for the group consisting of two same-sided cosets were given.

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What this paper is about

The concepts of one-sided coset and two-sided coset were given,some of properties of two-sided coset were discussed,and it was pointed out that two-sided coset b~(-1)Ha~(-1) is just consist of inverse elements of each element existing in two-sided coset aHb,and that aHb≤ Gb~(-1)Ha~(-1)≤G.With the use of the basic properties of one-sided coset and by use of the method of circulating proof,twenty six equivalent propositions on the groups consisting of two-sided coset aHb were proved.Finally,seven necessary and sufficient conditions for the group consisting of two same-sided cosets were given.

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Available abstract

The concepts of one-sided coset and two-sided coset were given,some of properties of two-sided coset were discussed,and it was pointed out that two-sided coset b~(-1)Ha~(-1) is just consist of inverse elements of each element existing in two-sided coset aHb,and that aHb≤ Gb~(-1)Ha~(-1)≤G.With the use of the basic properties of one-sided coset and by use of the method of circulating proof,twenty six equivalent propositions on the groups consisting of two-sided coset aHb were proved.Finally,seven necessary and sufficient conditions for the group consisting of two same-sided cosets were given.

Key concepts: Coset, Mathematics, Pure mathematics, Combinatorics, Inverse, Group (periodic table), Algebra over a field, Physics

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