Do you know natural/concrete/appealing examples of right/left cosets in group theory ?

This notion is a powerful tool but also a very abstract one for beginners so this is why I’m looking for friendly examples.

**Answer**

The plane R2 is a group under addition, and the x-axis {(a,0):a∈R} is a subgroup of it. Then the lines parallel to x-axis are precisely the cosets of this subgroup.

Instead of x-axis, you can take any line through origin; it will be a subgroup, and lines parallel to it will be cosets.

Similarly, C∗=C−{0} is group under multiplication; think it like a punctured plane. Then S1={z∈C:|z|=1} is a subgroup, which is a circle with center origin and radius 1. Its cosets are concentric circles to S1.

*Edit:* Consider the group GLn(k) of n×n invertible matrices over a field k and SLn(k) be the subgroup consisting of matrices with determinant 1. Then for every λ∈k−{0}, the subset of GLn(k) consisting of matrices with determinant λ is a coset of SLn(k) (where λ=1 gives trivial coset).

**Attribution***Source : Link , Question Author : projetmbc , Answer Author : p Groups*