Let N
be a normal subgroup of a group G.
The cosets of N
in G
form a group G/N
of order [G:N]
, 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.
To prove that the cosets of N in G form a group G/N of order [G:N], we need to show that the set of cosets satisfies the group axioms.
1. Closure: For any two cosets aN and bN, their product in G/N is defined as (aN)(bN) = abN. Since G is a group, ab is an element of G, and abN is a coset of N in G. Thus, the operation is closed.
2. Associativity: The operation of multiplication of cosets is associative since we are multiplying elements of the group G and then taking the coset of their product.
3. Identity: The identity element in G/N is the coset eN, where e is the identity element of G. For any coset aN, we have (aN)(eN) = aeN = aN = eNaN = (eN)(aN), showing that eN is the identity element.
4. Inverses: For any coset aN, its inverse in G/N is the coset a^-1N, where a^-1 is the inverse of a in G. This is because (aN)(a^-1N) = aa^-1N = eN = a^-1aN = (a^-1N)(aN).
Therefore, the set of cosets of N in G, under the operation of multiplication of cosets, forms a group G/N.
To show that the order of G/N is [G:N], we can use Lagrange's theorem. Lagrange's theorem states that for any subgroup H of a group G, the order of H divides the order of G. In this case, N is a subgroup of G, and [G:N] represents the index of N in G, which is the number of distinct cosets of N in G.
Since the distinct cosets of N in G partition G, the order of G can be expressed as [G:N] times the order of N. Therefore, the order of G/N is [G:N] because the order of G is [G:N] times the order of N.
Hence, the cosets of N in G form a group G/N of order [G:N].