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There is an isomorphism $$ \begin{align*} \varphi:G_{(p,q)}&\to\mathbb{Z}[1/q],\\ g_i&\mapsto\frac{p^i}{q^i}. \end{align*} $$ To check surjectivity: for any $\frac{a}{q^n}\in\mathbb{Z}[1/q]$, we can …

The multiplication does not give a Rng structure because it does not distribute over addition: $$ \pi\hat{\times}\big(\pi+\pi)=\pi \hat{\times} 0=0\neq \pi = \frac{\pi}{2}+\frac{\pi}{2}=\pi\hat{\times …

I don't think this is true. Let $G=F_2=\langle a,b\rangle$, the free group on two generators. Take $V=k^2$ (say $k$ a field of characteristic not 2), with $a$ acting trivially and $b$ acting by $(x,y …

The group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times \mathbb{Z}_p]]$ has the structure of a complete Hopf algebra, where $\mathbb{Z}_p\times \mathbb{Z}_p$ consists of precisely the group-like elements. …

The usual power series for $\log(1+x)$ determines an injection from $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$ into $\mathbb{Q}_p[[x_1,x_2]]$. A power series $f\in 1+(x_1,x_2)$ is in $\langle 1+x_1,1 …

The answer to the first question is no. Take as index set $I=\mathbb{N}$. Let $\varphi$ be the continuous automorphism of $V=\mathbb{Z}_p^{\mathbb{N}}$ given by $$ \varphi:(x_1,x_2,\ldots)\mapsto (x_1 …