17.18 (2010)

Let $\mathbf{A}$ be the class of compact groups $A$ with the property that whenever two compact groups $B$ and $C$ contain $A$, they can be embedded in a common compact group $D$ by embeddings agreeing on $A$. I showed (Manuscr. Math., 58 (1987) 253–281) that all members of $\mathbf{A}$ are (not necessarily connected) finite-dimensional compact Lie groups satisfying a strong “local simplicity” property, and that all finite groups do belong to $\mathbf{A}$.
$\qquad$ a) Is it true that $\mathbb{R}/\mathbb{Z} \in \mathbf{A}$?
$\qquad$ b) Do any nonabelian connected compact Lie groups belong to $\mathbf{A}$?
$\qquad$ c) If $A$ belongs to $\mathbf{A}$, must the connected component of the identity in $A$ belong to $\mathbf{A}$?