# Quotient group

• Apr 1st 2008, 06:55 PM
hercules
Quotient group
Please help me with the following proof....my mind doesn't seem to be working right now.

Assume N is a normal subgroup of G.
a) Prove that if [G:N] is a prime, then G/N is cyclic. (could you provide an example as well)
b) Prove or disprove the converse of the statement in part (a).

Thank You.
• Apr 1st 2008, 06:57 PM
ThePerfectHacker
Quote:

Originally Posted by hercules
Please help me with the following proof....my mind doesn't seem to be working right now.

Assume N is a normal subgroup of G.
a) Prove that if [G:N] is a prime, then G/N is cyclic. (could you provide an example as well)
b) Prove or disprove the converse of the statement in part (a).

a)If $G$ is a finite group then $|G/N| = [G:N]$. Prime groups are cyclic, so $G/N$ is cyclic.
b)It is false. Try something.
• Apr 1st 2008, 06:59 PM
hercules
Quote:

Originally Posted by ThePerfectHacker
a)If $G$ is a finite group then $|G/N| = [G:N]$. Prime groups are cyclic, so $G/N$ is cyclic.
b)It is false. Try something.

wow....lightening quick response ...thank you.
• Apr 1st 2008, 08:57 PM
ThePerfectHacker
Quote:

Originally Posted by hercules
wow....lightening quick response ...thank you.

I am the fastest responder in the West. (Emo)

Also, why would this Forum claim to be the fastest responding forum on the internet if it was not true?