Cyclic Groups and Cayley Diagrams in Mathematical Symbolism
...finitely many times returns to being the identity transformation, i.e: $f^n = \text{id}$. In this case the cyclic group $G$ generated by $f$ is a cyclic group of order $n$; that is, $G$ consists of the following $n$ elements: $\text{id}_D, f, f^2, f^3, \ldots, f^{n-1}$.
The second possibility is that $f^n \neq f^m$ for every $n \neq m$. In this case the cyclic group $G$ is a group of infinite elements; that is $G = \{f^n\}_{n \in \mathbb{Z}}$ where we set $f^0 = \text{id}_D$ and with $\mathbb{Z}$ we have denoted, as usual, the group given by all positive and negative integers with the operation of addition.
A suitable visualization for case 1) is that proposed above in the case of the binary law. In general, if we set $D := \mathbb{R}^2$, if with $f$ we visualize the rotation on the Euclidean plane $\mathbb{R}^2$ of an angle $\frac{2\pi}{n}$ centered at the origin and if we divide a circle of unit radius into $n$ equal arcs we have that considering the $n$-pointed star obtained from the $n$ points on the circle we obtain a geometric object that is invariant under the action of the finite cyclic group given by rotations of angle $\frac{2\pi k}{n}$, with $k = 0, 1, \ldots, n-1$.
We leave to the reader to intuit that in the second case it is possible to visualize $G$ having as invariant geometric object the circle.
Based on the preceding considerations we can therefore formulate the following law for the action of an isolated transformation:
The invariance under the action of any isolated transformation can be symbolically expressed through an n-pointed star or through the circle.
Naturally in the case of the action associated with the finite cyclic group we could also have taken the geometric object consisting of a regular polygon of $n$ sides inscribed, for example, in a circle. However such symbolization is closer to the passive aspect of substance compared to the active one of essence given by the $n$-pointed star.
To be able to transmit the basic principles for a full understanding, however, of all the principal transformations that act on a single domain $D$ we should indicate at least the salient points of the classification theorem of simple groups. We do not intend to do this in this study. We limit ourselves to recalling that simple groups are the equivalent of prime numbers in ordinary algebraic factorization: the study of every finite algebraic group can be reduced to the study of simple algebraic groups, which have recently been completely classified.
This great variety of spiritual realities is not surprising. The initiate, however, knows that, while infinite numbers exist, each with its own characteristics, from a symbolic point of view it is the understanding of the first 12 that constitutes the base for being able to undertake the initiatic way. In the same way, at least the knowledge of finite groups up to order 12 is the base for every conscious theurgical action.
Cayley diagrams. We now show how important symbols of the Tradition, such as the Seal of Solomon (also known as the Star of David), are interrelated to group theory. To understand such connections, we recall that given a group $G$ and one of its elements $g$, the Cayley diagram of the pair $(G,g)$ is the result of the action of $g$ applied to all the other elements.
Let us now consider the case of the cyclic group of order 6. This group can be realized by means of a rotation $r: \mathbb{R}^2 \to \mathbb{R}^2$ of 60°; that is $G = \{e, r, r^2, r^3, r^4, r^5\}$.
The Cayley diagrams of the group are therefore represented by the diagrams in Figure 5. In box 1 is represented the action of element $r$, or equivalently, but with reversed arrows, under the action of element $r^5$; in box 2 is represented the action of element $r^2$, or equivalently, but with reversed arrows, under the action of element $r^4$; in box 3 is represented the action of element $r^3$; in box 4 are reported the actions of the various elements simultaneously in which transpires the simultaneity of the action of the pole, the reaction of substance, the centripetal synthesis and the centrifugal effusion.
The symbolic study of group theory constitutes an important support for meditation. We observe that all groups satisfy an evident property for the operation of composition of automorphisms of a domain $D$. This operation is called the associative property of the product law. The meaning of this is that the composition of any three transformations $a, b, c$ belonging to the same group $G$ produces the same result whether one first composes $a$ with $b$ and then with $c$ or...