On mixed normal subgroups of the group Lim($\mathbb{N}$)

Let $\mathbb{N}$ be the set of natural numbers. A permutation $g$ of the set $\mathbb{N}$ is called limited if there exists $\alpha\in \mathbb{N}$ such that $|\beta-\beta^g|\leqslant|\alpha-\alpha^g|$ for every $\beta\in \mathbb{N}$. Denote by $G=\mathrm{Lim}(\mathbb{N})$ the group of all limited permutations of the set $\mathbb{N}$. In 2010 N. M. Suchkov and N. G. Suchkova proved that $G = AB$, where $A$ and $B$ are locally finite subgroups of $G$. In 2016 the same authors described the locally finite radical $R$ of the group $G$. In particular, the following result was proved: if $H$ is a normal subgroup of $G$, then either $H\leqslant R$ or $H$ is a mixed subgroup of $G$. In this paper we study mixed normal subgroups of the group $G$. It is proved that there exists a continuum set of such subgroups. We give examples of infinitely decreasing and infinitely increasing chains of mixed normal subgroups of $G$. In 2020 the authors proved similar results for locally finite normal subgroups of $G$.