# Normal subgroups

1. Mar 2, 2008

### ehrenfest

[SOLVED] normal subgroups

1. The problem statement, all variables and given/known data
My book states the following without any justification right before proving the Third Isomorphism Theorem: "If H and K are two normal subgroups of G and $K \leq H$, then H/K is a normal subgroup of G/K."
The elements of H/K are cosets of K in H. The elements of G/K are cosets of K in G. Therefore I think that statement is simply absurd. That is, the elements of H/K are not even contained in the quotient group G/K; therefore, they cannot possibly form a normal subgroup in G/K.

EDIT: wait, never mind, the cosets of K in H are also cosets of K in G; sorry
EDIT: and the reason H/K is normal in G/K is that gK*hK*g^(-1)K = (ghg^{-1})K = h' K since H is normal in G. Very EDIT: cool.

2. Relevant equations

3. The attempt at a solution

Last edited: Mar 2, 2008
2. Mar 2, 2008

### eok20

The elements in H/K are, indeed, contained in G/K since H is contained in G: A general element in H/K is hK which is in G/K since h is in G.