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Understanding a component of the secondary quadratic form
I am trying to understand the proof of the following fact:
There exists a number $c > 0$ such that any composition $n \in N$ is contained in the union of at most $c^n$ cosets of $Q$ and $c^{ -n}Q^{ -1}$.
Which is proved via induction on $n$ as follows:
Consider the group of invertible elements $A = GL_2(\mathbb{Z}/q\mathbb{Z})$ modulo the determinant $d := det(M)$ acting on the vector space $V = \mathbb{Z}^2$ via the given quadratic form (which I am not sure if it is standard notation in the math community) and let $B$ be the subgroup of the elements with determinant $1$.
Then since $B \cap Q = 1$ we can find a number $c > 0$ such that any composition $n \in N$ is contained in the union of at most $c^n$ cosets of $Q$ and $c^{ -n}Q^{ -1}$.
The part that I don’t understand is the last sentence. I would say that the fact the cosets are defined as all mappings $\phi:Q \to Q$ is that any composition is “determined” by the composition of any two elements but there is a number $c$ is the definition of a coset, so why do we need a number $c$?
Any help would be greatly appreciated, thanks.
A:
Let $H = \{x\in A\mid det(x)= 1\}$. There is an injective mapping $B\to H$. Let $c = max\{exp(log(x)\mid x\in H\}$. If $n\in B$, then $n^{ -1}= n^{\prime -1}$ and $n
eq 1$. Let $m = lcm\{exp(log(x)\mid n
eq x\in B\}$.
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