“Natural” example of cosets

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.


The plane R2 is a group under addition, and the x-axis {(a,0):aR} 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={zC:|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 λk{0}, the subset of GLn(k) consisting of matrices with determinant λ is a coset of SLn(k) (where λ=1 gives trivial coset).

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

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