Give an example of the desired subgroup and group if possibl

Give an example of the desired subgroup and group if possible. If impossible, say why it is impossible. A subgroup of an abelian group G whose left cossets and right cossets give different partitions of G A subgroup of a group G whose left cossets give a partition of G into just one cell A subgroup of a group of order 6 whose left cossets give a partition of the group into 6 cells A subgroup of a group of order 6 whose left cossets give a partition of the group into 12 cells

Solution

Consider Z₆

Z₆ is agroup of order 6.

{0} is a subgroup of Z₆ whose left cosets give a partition of Z₆ into 6 cells.

Ans : {0} is the example for the subgroup and group is Z₆