# Finding the cyclic subgroup...

• Oct 21st 2009, 11:10 AM
elninio
Finding the cyclic subgroup...
Find the cyclic subgroup generated by
2 1
0 2
(matrix)
in $\displaystyle GL_2(Z_3)$.

I suppose im confused as to what exactly it's asking me to do.
$\displaystyle GL_2(Z_3)$ signifies 2x2 matrices with mod3 entried, correct? How do I find the cyclic subgroup?
• Oct 21st 2009, 11:15 AM
Swlabr
Quote:

Originally Posted by elninio
Find the cyclic subgroup generated by
2 1
0 2
(matrix)
in $\displaystyle GL_2(Z_3)$.

I suppose im confused as to what exactly it's asking me to do.
$\displaystyle GL_2(Z_3)$ signifies 2x2 matrices with mod3 entried, correct? How do I find the cyclic subgroup?

The cyclic subgroup generated by the element $\displaystyle a$ is all elements of the form $\displaystyle a^n$ for $\displaystyle n \in \mathbb{N}$. So, just find $\displaystyle A$, $\displaystyle A^2$, $\displaystyle A^3$, etc until you get back to the identity. The powers of your generator will be your group.
• Oct 21st 2009, 11:18 AM
elninio
Thank you! I've found the answer. Its too long to write out but it contains six matrices.
• Oct 21st 2009, 12:00 PM
tonio
Quote:

Originally Posted by elninio
Thank you! I've found the answer. Its too long to write out but it contains six matrices.

Correct.

Tonio