Let $H=\left\{ z\in\mathbb{C}\mid\Im\left(z\right)>0\right\}$ be the upper-half Poincare plane. Let $GL\left(2,\mathbb{R}\right)$ be the general linear group, $Z\left(GL\left(2,\mathbb{R}\right)\right)$ be the center of the general linear group and $O\left(2,\mathbb{R}\right)$ be the orthogonal subgroup of $GL\left(2,\mathbb{R}\right)$.
What does it mean to say $H=GL\left(2,\mathbb{R}\right)/\left(Z\left(GL\left(2,\mathbb{R}\right)\right)\cdot O\left(2,\mathbb{R}\right)\right)$? The left-hand side is a metric space and the right hand side is a set of cosets of $GL\left(2,\mathbb{R}\right)$. So I'm confused about what it means to write that they are equal or to say "the upper half plane is..." It seems like this would be the group of orientation preserving isometries of H, but I still find the terminology confusing.
I've been trying to figure out what this could possibly mean, but my searches on the internet have not been fruitful. I've also looked at 2 sources on standard modular groups but they make no mention of this fact. An explanation or reference would be greatly appreciated.
Motivation: I am reading a paper titled "On Modular Functions in characteristic p" by Wen-Ch'ing Winnie Li which can be found at http://www.jstor.org/stable/1997973. The claim appears on page 3 of the pdf (page 232 of the journal). It is also stated on the wikipedia page: http://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model