This is from Abstract Algebra Show that a subgroup N of a gr

This is from Abstract Algebra:

Show that a subgroup N of a group G is normal if and only if each right coset Nx of N with an element x of G is equal to the left coset yN with some element y of G.

I know I want to show Nx = yN but not really sure how to prove it. My teacher gave me the hint about assuming N is normal (as a subgroup), we then have to show that for any x, there\'s a y with Nx=yN.

Solution

A subgroup N of G is normal if it satisfies (any of) the following equivalent conditions:

1. every left coset of N is a right coset;

2. gN = Ng for every g G;

3. gNg1 = N for every g G, where gNg1 = {gng1 | n N}.

If N is a normal subgroup of G then we write N E G. If N E G then we can define a group structure on the set of cosets of N in G by (g1N,g2N) = g1g2N. The resulting group is called the quotient of G by N, and denoted G/N. There is no such quotient defined for subgroups which are not normal.

proof

Let G be a group and N be a sub group N less than or equal to G .If a left coset gN is equal to that Ng\' then,since 1 N ,we know that g Ng\'.If g Ng\' then Ng\'=Ng as right coset .Hence that creterion that a subgroup is such that every left coset is equals to right coset can be summerised in the following definition.

definition:

Let G be a group. A subgroup N < G is called normal if gN=Ng for all g G.