I am confused by a question, which is probably of school level.
In some papers I have seen an induction from the group $GL_2(\mathbb{Z}_p)\cdot\mathbb{Q}_p^\times$ to the group $GL_2(\mathbb{Q}_p)$, where $\mathbb{Q}_p$ is the field of p-adic numbers and $\mathbb{Z}_p$ is its ring of integers. My confusion is that I cannot understand how these two groups are different.
I understand that $GL_2(\mathbb{Q}_p)/GL_2(\mathbb{Z}_p)$ is the set of 2-dimentional $\mathbb{Z}_p$-lattices and that $GL_2(\mathbb{Q}_p)/\mathbb{Q}_p^\times\cong PGL_1(\mathbb{Q}_p)$, but my confusion still remains.
Direct question: is there an element of $GL_2(\mathbb{Q}_p)$ that cannot be presented as a product of an element of $GL_2(\mathbb{Z}_p)$ and nonzero p-adic number?
As noted in comments, $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$ is such a matrix.
It might be worth noting that the quotient $\mathrm{GL}_2(\mathbb Q_p)/\mathrm{GL}_2(\mathbb Z_p)\mathbb Q_p^{\times}$ is an infinite set, which is often conveniently represented as the set of vertices of a regular tree of valence $p+1$.
In this representation, the set of cosets of the matrices $\begin{pmatrix} p^n & 0 \\ 0 & 1\end{pmatrix}$ for $n \in \mathbb Z$ then form a line in this tree (infinite in both directions).