Structure of hierarchical linear systems with cyclic symmetry

In this paper, we analyze the structure of hierarchical linear systems that are invariant under actions of finite cyclic groups. In particular, we consider N = n 1 n 2 ⋯ n m identical d -dimensional linear systems grouped into m hierarchies, such that in each level of hierarchy the partitioned subsystems are invariant under the action of cyclic groups. We prove that the equivalence classes of all possible partitions of such hierarchical systems have one–one correspondence with the decomposition of N th order abelian groups into cyclic groups of prime power order.