closed normal subgroup of $ G $ which we will define in .

\[Intro trichotomy\] [(i)]{} The case [[$ ({\bf C1}) $]{}]{} occurs iff $ H_\mu(\bP^o) = G $ for some (and hence any) extremal point $ \bP^o $.\
[(ii)]{} The case [[$ ({\bf C2}) $]{}]{} occurs iff $ H_\mu^s = {\left\{ 0 \right\}} $. If it is true, then, among the solutions, all extremal points are strong and the others non-strong.\
[(iii)]{} The case [[$ ({\bf C3}) $]{}]{} occurs iff $ H_\mu(\bP^o) \subsetneq G $ and $ H_\mu^s \supsetneq {\left\{ 0 \right\}} $.

Theorem \[Intro trichotomy\] will be proved in section \[sec: trichotomy\].

The subgroups $ H_\mu^s $ and $ H_\mu(\bP^o) $ for extremal points $ \bP^o $ are related as follows.

\[Intro isot group\] For any extremal point $ \bP^o $, the inclusion $$H_\mu(\bP^o) \subset H_\mu^s$$ holds. If $ G $ is abelian, then the equality holds for any extremal point $ \bP^o $.

We will restate Theorem \[Intro isot group\] as Theorem \[thm: isot group\] including further information. In the case where $ G = \T = \R/\Z $, note that the abelian group $ \Z $ is the dual group of $ G $ in the 