Example 4

Similarly, let N be a finite abelian group, and $\varphi :\pi_1(G)\rightarrow N$ a surjective homomorphism. Let $G'$ be the covering of G associated to this homomorphism; this is a connected compact Lie group, of which N is a central subgroup (General Topology, Chap. XI, in preparation), and G can be naturally identified with $G'/N$. Let $T'$ be the maximal torus of $G'$ that is the inverse image of T. If $(P,P_0,S)$ is the covariant diagram of G relative to T, the covariant diagram of $G'$ relative to $T'$ can be identified with $(P',P'_0,S)$, where $P'$ is the kernel of the composite homomorphism $\varphi \circ f(G,T) : P\rightarrow N$ (cf. no. 6, Prop. 11), and $P'_0= P_0\cap P'$.