# Subgrounds of Symmetric Groups

• Nov 9th 2008, 01:28 PM
apsis
Subgrounds of Symmetric Groups
I'm trying to find all of the subgroups of the symmetric $\displaystyle S_{3}$. But am not sure how to generate these groups as all of the properties of groups must be satisfied.

I am thinking it is something along the lines of:
{(123)->(123)} (no change)
{(123)->(123), (312), (231)}
{(123)->(123),(132), (213), (321)}

Is this the right track? Or am out out in left field?
• Nov 10th 2008, 08:11 PM
ThePerfectHacker
Quote:

Originally Posted by apsis
I'm trying to find all of the subgroups of the symmetric $\displaystyle S_{3}$. But am not sure how to generate these groups as all of the properties of groups must be satisfied.

I am thinking it is something along the lines of:
{(123)->(123)} (no change)
{(123)->(123), (312), (231)}
{(123)->(123),(132), (213), (321)}

Is this the right track? Or am out out in left field?

The subgroups are $\displaystyle \{ e \}, \left< (12) \right>, \left< (13)\right>, \left< (23) \right>, \left< (123) \right>,S_3$.