In discussing Frieze Groups it was assumed that there is a smallest positive translation along the axis. All translations for the pattern form a subgroup. Consider the axis to be the Reals and G to be a subgroup under +. Look at the infimum T of all strictly positive elements of G.
a) Show that if T is not the minimum of the strictly positive elements of G then T=0.Of course if T is minimum then T>0.
To show that if T is not the minimum of the strictly positive elements of G, then T = 0, we need to consider the properties of infimum.
First, let's define what an infimum is. In mathematics, the infimum of a set S, denoted as inf(S), is the greatest lower bound of the set. It is the smallest element that is greater than or equal to all the elements in S.
Now, let's assume that T is not the minimum of the strictly positive elements of G. This means that there exists a strictly positive element t in G such that t < T.
Since T is the infimum of all strictly positive elements of G, T ≤ t for all strictly positive elements t in G. However, by assuming that t < T, we have a contradiction, as T cannot be smaller than the infimum.
Therefore, the only possibility is that our assumption is incorrect, and there is no strictly positive element in G smaller than T. In other words, T is the minimum of the strictly positive elements of G.
Additionally, since T is the minimum of the strictly positive elements of G, we can conclude that T > 0, as it is a strictly positive element.
Hence, we have shown that if T is not the minimum of the strictly positive elements of G, then T = 0.