Groups that Distribute over n-Stars

Suppose (S, ∗) is a mathematical structure on a set S. As examples, (S, ∗) might be a topological space on S, a topological group on S, an n-ary operator on S, an n-ary relation on S or a Steiner triple system on S. A similarity mapping on (S, ∗) is a permutation on S that preserves the structure of (S, ∗). Such mappings f : (S, ∗) → (S, ∗) are called by different names. As examples, f might be called a homeomorphism or an automorphism on (S, ∗). See [5] for the details. Suppose (S, ·) is a group on S. For each fixed t in S, the left and right translation on S are the permutations Lt (x) = t · x and Rt (x) = x · t respectively. They are called the left and right translation by t. See [1] and [5]. We say that the group (S, ·) left-distributes or right-distributes over (S, ∗) if respectively for all t in S,Lt (x) : (S, ∗) → (S, ∗) or Rt (x) : (S, ∗) → (S, ∗) is a similarity mapping on (S, ∗). Naturally occurring examples of this phenomenon are rare. One such example is a topological group in which a group both left-distributes and right-distributes over a topological space. See [5] for more examples. In this paper, we give a naturally occurring example that involves a double 7 pointed star which is structurally the same as 7 lines in the plane intersecting in C7 2 = 21 distinct points. However, the overwhelming main purpose of this paper is to point out the question of the existence of a group (S, ·) that left-distributes or right-distributes over a given structure (S, ∗). The authors hope this paper will lead to some undergraduate research. 174 H. Reiter and A. Holshouser