(adsbygoogle = window.adsbygoogle || []).push({}); [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 [itex]K \leq H[/itex], 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Normal subgroups

**Physics Forums | Science Articles, Homework Help, Discussion**