is divided into two cosets.One is left coset represented by and the other is right coset represented by.

It seems like your question pertains to cosets in group theory, where a group can be divided into left and right cosets with respect to a subgroup.

Explanation of Cosets

1. Left Cosets: Given a group $G$ and a subgroup $H$, the left cosets are defined as the set of elements obtained by multiplying each element of $H$ by a fixed element $g$ in $G$ from the left. Mathematically, the left coset of $H$ in $G$ with representative $g$ is given by: $gH = \{ gh : h \in H \}$

2. Right Cosets: Similarly, the right cosets are defined by multiplying each element of $H$ by a fixed element $g$ in $G$ from the right. The right coset of $H$ in $G$ with representative $g$ is given by: $Hg = \{ hg : h \in H \}$

In group theory, the left and right cosets can be different, especially in non-abelian groups. In abelian groups, every left coset is equal to the corresponding right coset, meaning they are inherently identical.

Summary

To conclude, cosets allow us to partition a group into distinct subsets under the operation defined in the group. Understanding the left and right cosets' structures is crucial for higher-level concepts in group theory, including analyzing group actions, determining normal subgroups, and studying quotient groups. If you have specific groups or examples in mind, feel free to share them for more detailed analysis!