Is there a standard notation for the multiplicative group generated by the primes $p\in P$? Is there a standard notation for the multiplicative group generated by the primes $p\in P$?
Let $P$ be some set of primes e.g. $P=\{2,3\}$
Then $G_P$ is the multiplicative group generated by these primes so e.g. $G_{\{2,3\}}$ is the 3-smooth numbers and their inverses, with multiplication.
Is there a standard notation or way of expressing this group and similar?

When $P = \{2, 3\}$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j \in \Bbb{Z}$ and this is easily checked to give an isomorphism between $G_P$ and the sum $\Bbb{Z}^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality $|P|$ of $P$, so one standard notation for $G_P$ is $\Bbb{~Z}^{|P|}$.
This should be understood subject to the proviso that, if $P$ is infinite, $\Bbb{~Z}^{|P|}$ is to be interpreted as the infinite sum and not the infinite product (i.e., it only includes sequences $(i_1, i_2, \ldots)$ where all but finitely many of the $i_j$ are zero).
