Kaluzhnin's representations of Sylow $p$-subgroups of automorphism groups of $p$-adic rooted trees

The paper concerns the Sylow $p$-subgroups of automorphism groups of level homogeneous rooted trees. We recall and summarize the results obtained by L. Kaluzhnin on the structure of Sylow $p$-subgroups of isometry groups of ultrametric Cantor $p$-spaces in terms of automorphism groups of rooted trees. Most of the paper should be viewed as a systematic topical survey, however we include some new ideas in last sections.