Clearly all bijections functions $\varphi : \mathbb N \to \mathbb N$ form a group, let's call it $S(\mathbb N)$. Now let's define $H \subset S(\mathbb N)$ to be the set of all elements $\varphi \in S(\mathbb N)$ such that for every real valued sequence $(a_n)_{n \in \mathbb N}$ for which $$\sum_{n=1}^\infty a_n$$ converges we have $$\sum_{n=1}^\infty a_n = \sum_{n=1}^\infty a_{\varphi(n)}.$$ Clearly $H$ forms a proper subgroup of $S(\mathbb N)$. For fixed $n \in \mathbb N$ the group $S_n$ is a subgroup of $H$. But this is not everything. For example the map $$ \varphi(n) := n+(-1)^{n+1} $$ is also in $H$ without being a finite permuation.
Question 1: Does the group $H$ have a special name? Is it mentioned somewhere in literature?
Question 2: Is $H$ simply the group of all "infinite products of disjoint cycles"?
