If A denotes the union of these cosets formed by the elements of a subgroup of the quotient group H, show that A is a subgroup of G.

Question

I'm having trouble with this problem.

Answer 1

You need to show that

$$A:=\{x\in G\;:\;xN\in H(\le G/N)\}\le G$$

For example:

$$H\le G/N\implies\;\text{the unit of the quotient is in}\;\;H\implies 1N=N\in H\implies 1\in A$$

$$x,y\in A\implies xN,yN\in H\implies xNyN:=(xy)N\in H\implies xy\in A$$

Can you take it from here? And you can do more: show that

$$\;H\lhd G/N\iff A\lhd G\;\;,\;\;\text{and also}\;\;[G/N:H]=[G:A]$$

The above is the wonderful and very helpful Correspondence Theorem for groups.