# Quotient groups of cyclic groups

• Jul 21st 2013, 02:38 PM
phys251
Quotient groups of cyclic groups
"G is a cyclic group with normal subgroup N. Show that G/N is cyclic."

Not sure what to do here. I know that all subgroups of a cyclic group are also cyclic. But the quotient group is confusing me here.
• Jul 21st 2013, 10:06 PM
chiro
Re: Quotient groups of cyclic groups
Hey phys251.

Check out this:

Cyclic group - Wikipedia, the free encyclopedia

and look at the property regarding prime numbers for the order of the groups
• Jul 23rd 2013, 06:48 PM
phys251
Re: Quotient groups of cyclic groups
OK thanks, I'll look into that. I just need to get more familiar with the definition of quotient groups.
• Jul 30th 2013, 04:31 PM
Deveno
Re: Quotient groups of cyclic groups
First of all, note that cyclic groups are of necessity abelian. So ANY subgroup is automatically normal.

Since G is cyclic, there is some element g of G that generates G. I claim gN generates G/N.

For suppose xN is ANY element of G/N. Since x is some element of G, we have $x = g^k$, for some integer k (since G is cyclic, with generator g).

Thus: $xN = g^kN = (gN)^k$, which shows that G/N is generated by gN.