A coset in abstract algebra is a subset of a group that is obtained by multiplying a fixed element of the group by all elements of a subgroup.
To prove that two cosets are equal, you need to show that they have the same elements. This can be done by showing that every element in one coset can be obtained by multiplying an element in the other coset by an element in the subgroup.
Yes, a coset can contain more than one element. In fact, a coset must have the same number of elements as the subgroup that it is generated by.
In abstract algebra, a left coset is a coset obtained by multiplying a fixed element of a group on the left by all elements of a subgroup. A right coset is a coset obtained by multiplying a fixed element of a group on the right by all elements of a subgroup.
Cosets are an important concept in group theory as they help us understand the structure of a group and its subgroups. They also help us prove important theorems such as Lagrange's theorem and the first isomorphism theorem.