# Thread: Finding the cyclic subgroup...

1. ## 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?

2. 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.

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

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

Correct.

Tonio

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### how do we find the cyclic subgroups generated by a matrix

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