Homomorphic encoders of profinite abelian groups I

Abstract

Let {G i : i ∈ N} be a family of finite Abelian groups. We say that a subgroup G ≤and order(c 1 ) divides order (c |[1,ni] ). In this paper we investigate the structure of order controllable subgroups. It is known that each order controllable profinite abelian group is topologically isomorphic to a direct product of cyclic groups (see [8,15]). Here we improve this result and prove that under mild conditions an order controllable group G contains a set {g n : n ∈ N} that topologically generates G, and whose elements g n have all finite support. As a consequence, we obtain that if G is an order controllable, shift invariant, group code over an abelian group H, then G possesses a canonical generator set. Furthermore, our construction also yields that G is algebraically conjugate to a full group shift. Some connections to coding theory are also highlighted.

Homomorphic Encoders of Profinite Abelian Groups II

“…Now, repeating the same arguments as in Theorem 3.2 in [2], it follows that ) . Furthermore, using the σ-invariance of G, we can extend Φ 0 canonically to continuous onto group homomorphism…”

“…In this section, we apply the result accomplished in Theorem 3.2 in [2] in order to prove that the order-controllable group shifts over a finite abelian group possess canonical generating sets. Furthermore, our construction also yields that they are algebraically conjugate to a full group shift.…”

“…This article focuses on the research about (topological) groups that can be embedded into a product of finite groups, started in [1][2][3] (for a nice elementary example, consider the Rubik's cube group; every rotation provides a transformation on angles and edges and therefore, the Rubik's cube group can be embedded in a direct product (see http:// sporadic.stanford.edu/bump/match/rubik.html, accessed on 1 March 2022)). In particular, we deal here with the algebraic structure of abelian group codes.…”

“…Proof. (1) Using that G and G[p] are weakly controllable, we can proceed as in Theorem 3.2 in [2] in order to define a subset…”

“…Now, taking into account that lim n→±∞ σ n (y j ) = 0 for all 1 ≤ j ≤ m, we proceed as in Theorem 3.2 in [2] in order to lift Φ to a continuous group homomorphism…”