Let N

Question

be a normal subgroup of a group G.
The cosets of N
in G
form a group G/N
of order [G:N]

Answer 1

, called the quotient group or factor group. The union of all cosets of N in G is the set G itself. The product of two cosets aN and bN is defined as (ab)N, where ab is the usual product in the group G. The identity element of G/N is the coset N, and the inverse of a coset aN is the coset a^{-1}N. The quotient group inherits many properties from the original group G, such as the existence of subgroups and homomorphisms. It is an important tool in the study of group theory and has applications in algebraic geometry, cryptography, and other areas of mathematics.