I need to prove that H={(1 n 0 1) | n is an element of the integers} is a cyclic subgroup of GL2(reals). H is a matrix with the first row 1 and n and the second row 0 and 1. I appreciate the help!

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- Jan 30th 2013, 06:35 PMlovesmathCyclic Subgroup
I need to prove that H={(1 n 0 1) | n is an element of the integers} is a cyclic subgroup of GL2(reals). H is a matrix with the first row 1 and n and the second row 0 and 1. I appreciate the help!

- Jan 30th 2013, 07:41 PMDevenoRe: Cyclic Subgroup
i suppose you mean:

$\displaystyle H = \left\{\begin{bmatrix}1&n\\0&1 \end{bmatrix} | n \in \mathbb{Z} \right\}$

the way to do this is show that:

$\displaystyle \phi:H \to \mathbb{Z}$ defined by:

$\displaystyle \phi\left(\begin{bmatrix}1&n\\0&1 \end{bmatrix}\right) = n$

is an isomorphism of groups. this means that if:

$\displaystyle A = \begin{bmatrix}1&k\\0&1 \end{bmatrix};\ B = \begin{bmatrix}1&m\\0&1 \end{bmatrix}$

you need to show that:

$\displaystyle \phi(AB) = \phi(A) + \phi(B)$

and that $\displaystyle \phi$ is bijective.