<p><b>Abstract</b>—This paper presents a new decomposition technique for hierarchical Cayley graphs. This technique yields a very easy implementation of the divide and conquer paradigm for some problems on very complex architectures as the <it>star graph</it> or the <it>pancake</it>. As applications, we introduce algorithms for broadcasting and prefix-like operations that improve the best known bounds for these problems. We also give the first nontrivial optimal gossiping algorithms for these networks. In star-graphs and pancakes with <it>N</it> = <it>n</it>! processors, our algorithms take less than <tmath>$\lceil {\rm log}\ N\rceil + 1.5n$</tmath> steps.</p>