Group Theory: Commutativity and Geometric Realizations

it composes $a$ with the composition of $b$ and $c$; in formulas: $a(bc) = (ab)c$. Although an in-depth study about the extension of the associative property is quite relevant for the initiatic path, here we limit ourselves to an important observation about another important property that is not satisfied by all groups: the commutative property. Its meaning is that the order with which the group operations are performed does not change the result, that is, for every element $a, b$ in $G$ it holds that $ab = ba$. Indeed, one of the characterizing traits of all cyclic groups is precisely the commutativity of the operation. In general, the commutative property is not always satisfied. The scenario is where such important dichotomy emerges. In fact, it is easy to demonstrate that all groups of order less than 6 are commutative, that is, for them the commutative property holds, while there exist only two groups with 6 elements. However, while the first of these groups is the cyclic group $C_6$ of which we have previously described a geometric realization that is certainly commutative, the other group of order 6 is non-commutative. The latter is given by the group of permutations of 3 elements, often indicated with the following symbol: $S_3$. Figure 6: Cayley diagrams of group $S_3$. In panel 1 the action of element $r$ is represented, or equivalently but with reversed arrows under the action of element $r^2$; in panel 2 the action of element $f$ is represented; in panel 3 the action of both generators is represented. In the first of these groups, namely $C_6$, we have a unique generator, which we had indicated in our realization as $r$, while in the second case we have two distinct generators, which we will indicate as $r$ and $f$. Geometrically we can realize $r$ as a rotation of $120° = 2\pi/3$ and the transformation $f$ as a reflection with respect to a mirror. The group of permutations of 3 elements turns out to be formed by the following elements: $S_3 = \{e, r, r^2, f, rf, r^2f\}$ where $fr \neq rf$ and, instead, it holds that $fr = r^2f$. The Cayley diagrams of $S_3$ represented in Figure 6 allow us to clarify its meaning and understand the symbolic distinction that exists between the two groups of order 6. In the case of a cyclic group, the elements always have the same origin and are of the same degree of reality; elements that are powers of the same generator always commute with each other. They can exchange with one another in their operation, since they belong to the same hierarchical order. This is the case of the elements of cyclic group $C_6$. In the case of group $S_3$, the elements do not always commute with each other and, not by chance, there does not exist a single generator. The elements can be grouped into two subsets, one given as follows: $\{e, r, r^2\}$ and the other as follows: $\{f, rf, r^2f\}$. We observe that $\{f, rf, r^2f\} = \{f, fr, fr^2\}$ even though $fr \neq rf$, since $fr^2 = rf$ and $fr = r^2f$. The graphic realization of this distinction we find in the first Cayley diagram of $S_3$, Figure 6.1. Such diagram originates two Ternaries, one in front of the other, not equal nor interchangeable, but specular. They symbolize the human Trinity in its evolution of Power, Wisdom and Love, in accord and harmony with the Divine Trinity manifesting itself as Omnipotent, Omniscient and All-loving. The two Ternaries are similar, but of different substance: the first is symbol of individual substance, the second of uncreated Personality and of Divine essence. The link between the two Trinities is expressed by the specular operation $f$: the two Trinities mirror themselves one in the other, the contemplation of the Trinity that sees itself in the other. Music, Platonic Solids and Sacred Triangle (Solis) The corporeal manifestation, as immediately perceived with external sense, seems to evolve in a world with only three spatial dimensions. If, in particular, we conceive matter as completely and uniquely contained in it, then it will be seen to be subject to strong structural limitations. Indeed, the geometric-mathematical laws that govern the three-dimensional space of the external world greatly limit its possible spatial configurations. An important theorem of group geometry demonstrates that only 32 types of possible crystalline structures of matter can be reduced. Such modes are determined by the 32 types of possible point crystallographic groups that, precisely, govern the possible crystalline distributions of matter. The existence and nature of constraints of geometric nature about the deep structure of corporeal manifestation has always been considered an important support for metaphysical meditation. In traditional metaphysical conception, the solids commonly called "Platonic," also known by the name of "Cosmic Figures" in certain Pythagorean circles, are supports for meditation on some subtle properties of spiritual realities, which manifest in various forms.