Ring $R_{ 188 }$

Description:

Consider the binary representations of nonnegative integers. Let $\sigma_i$: $\Bbb N \to \Bbb N$ be the mapping that changes the $i$-th bit in the representations. The set $\{\sigma_i: i = 0, 1, \ldots\}$ generates a countable Abelian group $H$. Take a set $\Lambda$ of cardinality larger than continuum and form the direct sum $A = \bigoplus\limits_{i \in \Bbb N, \,\lambda \in \Lambda} A_{i\lambda}$, where $A_{i\lambda} = \langle a_{i\lambda}\rangle$ are cyclic groups of order $2$. For each $h \in H$, the map $a_{i\lambda} \mapsto a_{h(i) \lambda}$ extends to an automorphism $\varphi_h$ of the group $A$. The ring is $\Bbb QG$, where $G$ is the semidirect product $A \rtimes H$.

Notes: $G$ here is a locally finite solvable group of derived length 2

Keywords group ring