Cosets in a factor group are subsets of a group that are formed by multiplying a fixed element of the group by all the elements of a subgroup.
Normal subgroups are a special case of cosets, where the cosets are identical to the subgroup itself. In other words, all cosets in a normal subgroup are equal.
Cosets play a crucial role in group theory as they help to partition a group into smaller, more manageable subsets. This allows for the study of the structure and properties of a group in a more systematic way.
Yes, the number of distinct cosets of a subgroup in a group is equal to the index of the subgroup, which in turn is a factor of the order of the group. Therefore, cosets can be used to determine the order of a group.
If all the left cosets of a subgroup are equal to its right cosets, then the subgroup is normal. This is known as the "normal subgroup test" and is a commonly used method to determine the normality of a subgroup.