Corollary

If N satisfies the conditions of b) (iv), and if $K_1$ and $K_2$ are two maximal compact subgroups of G, there exists $n\in$ N such that $nK_1n^{-1}= K_2$.

Indeed, by (ii) there exists an element $g\in G$ such that $gK_1g^{-1}= K_2$; by (iv), there exists $n\in N$ and $k\in K_1$ such that $g=nk$. The element $n$ then has the required properties.