What is a group?

The New GROUP of World Servers:
What is a GROUP?

Summary: Every Service Unit within the New Group of World Servers operates as a GROUP, embodying an inner structure that is mathematical and fractal. It mirrors the TRIANGLE, the envisioned outer planetary etheric body for this cycle. Each unit amplifies its heart and mind threefold (2^3=8), shaping itself into an eight-pronged (cubic) living cell within the overarching organism. At the core of each unit lies the Zero – the Identity element – the servant leader.

A servant leader is one who shares power, puts the needs of others first, and helps people develop and perform as highly as possible. Just like the identity element in a group doesn’t change the other elements but enables them to maintain their properties, a servant leader doesn’t take over but facilitates the team members to achieve their best. -JB

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These new groups are appearing everywhere all over the world. The groups upon the outer plane, with their diversity of names and stated aims, are not connected with this inner group which is sponsoring or “projecting” the new groups, except in so far as they have a definite, even if nebulous, connection. This becomes always possible where there are three members of the New Group of World Servers found in any one exoteric group; it then becomes “linked by a [Page 197] triple thread of golden light” to the New Group of World Servers, and can in some measure be used. This great and spiritual grouping of servers is, on the physical plane, only very loosely linked. On the astral plane the linking is stronger and is based upon love of humanity; on the mental plane the major linking takes place, from the angle of the three worlds as a whole. It will be apparent, therefore, that certain developments must have taken place in the individual before he can consciously become a functioning member of the New Group of World Servers, which is the principal group at this time definitely working under the Law of Group Progress.

He must have the heart center awakened, and be so outgoing in his “behavior” that the heart is rapidly linked up with the heart centers of at least eight other people. Groups of nine awakened aspirants can then be occultly absorbed in the heart center of the planetary Logos. Through it, His life can flow and the group members can contribute their quota of energy to the life influences circulating throughout His body. The above piece of information is only of interest to those who are spiritually awakened, and will mean little or nothing to those who are asleep.

Esoteric Psychology, Vol. II by Alice A. Bailey

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A group is a fundamental concept in abstract algebra, used to model mathematical structures and operations. A group is a set equipped with an operation that combines any two of its elements to form a third element satisfying four conditions or axioms:

1. Closure: If ‘a’ and ‘b’ are elements in the group, then the result of the operation, say ‘a * b’, is also in the group.
2. Associativity: For all ‘a’, ‘b’, and ‘c’ in the group, the equation ‘(a * b) * c = a * (b * c)’ holds.
3. Identity element: There is an element ‘e’ in the group such that for every element ‘a’ in the group, the equations ‘e * a’ and ‘a * e’ both return ‘a’.
4. Inverse element: For each element ‘a’ in the group, there exists an element ‘b’ in the group, commonly denoted ‘a^(-1)’, such that ‘a * b = b * a = e’, the identity element of the group.

For example, consider the set of integers under addition. This forms a group because it satisfies all four axioms: the sum of any two integers is an integer (closure), the associative law holds for integers (associativity), zero serves as the identity element (any number plus zero equals the number itself), and every integer has an inverse (the negative of the number) that when added to the number gives zero (inverse).

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While abstract algebra and human social structures are very different fields, it’s possible to draw some conceptual similarities.

Let’s consider the four group axioms:

1. Closure: In a team, this could be interpreted as the idea that any combination of team members should still form a part of the team. This is crucial for collaboration, as sub-teams often need to form to tackle specific tasks.
2. Associativity: This might be seen in the way that team members can collaborate. For example, if Alice, Bob, and Charlie are working on a project, it shouldn’t matter if Alice first talks to Bob and then they both talk to Charlie, or if Alice first discusses with Charlie and then they both approach Bob. The end result should be the same.
3. Identity element: In a group context, an identity element could be a person or rule that, when added, doesn’t change the dynamic of the group. This could be a facilitator or a ground rule that helps the group function without altering its fundamental dynamics.
4. Inverse element: This is where the analogy comes into play. In a team, a person could serve as the “inverse” of another if their strengths compensate for the other’s weaknesses, and vice versa. This doesn’t mean they negate each other, but rather that they balance each other out. For example, in a team, you might have someone who is very detail-oriented (a trait beneficial for catching errors), and someone else who is more big-picture focused (great for strategic planning). These two team members could serve as inverses, their combined strengths making the team more effective.

It’s important to remember that these are metaphorical comparisons and real-life groups of people are much more complex and nuanced than mathematical groups. However, thinking in these terms might provide some interesting insights into how teams and groups can function more effectively.

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In the mathematical theory of groups, there isn’t a direct equivalent to the concept of leadership because all elements in a group are treated equally. However, if we consider the axioms metaphorically in the context of human groups, we could frame the role of leadership within these axioms.

The identity element could be seen as a servant leader. A servant leader is one who shares power, puts the needs of others first, and helps people develop and perform as highly as possible. Just like the identity element in a group doesn’t change the other elements but enables them to maintain their properties, a servant leader doesn’t take over but facilitates the team members to achieve their best.

Alternatively, a leader might be thought of as an operator. In a mathematical group, an operator is what combines two elements. In a team, a leader often brings together different team members, harmonizing their abilities towards a common goal.

A leader could also act as a homomorphism, mapping the group’s internal dynamics to an external context (such as organizational goals or broader societal contexts) and vice versa. They would ensure that the group’s internal operations align with external expectations and needs, preserving the structure and function of the group.

A group homomorphism is a way to translate one group into another while preserving the group structure. It’s a bit like translating a poem from one language to another, trying to keep not just the meaning of the words, but also the rhyme and rhythm of the original.

It’s important to note that these are metaphors, and human leadership is a complex field with many theories and models. These comparisons can provide some interesting food for thought, but they should not be taken as a complete or literal translation of leadership concepts into the language of abstract algebra.

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Modular arithmetic is a way of counting or doing calculations that “wraps around” after reaching a certain number. It’s like a clock that goes back to zero after reaching 12 (noon or midnight).

Let’s take the example of using modulo 12 for representing time on a 24-hour clock. In a 24-hour day, we count the hours from 0 to 23. However, if we want to display the time in a more familiar 12-hour format, we can use modular arithmetic with modulo 12.

When we apply modulo 12 to a number, it gives us the remainder when that number is divided by 12. For example, 14 modulo 12 is 2 because 14 divided by 12 equals 1 with a remainder of 2.

In the context of time, if we have 17:00 (5 PM) as our input, we can convert it to modulo 12 by taking the hours part (17) modulo 12, which gives us 5. So, 17:00 is equivalent to 5:00 in a 12-hour clock.

This approach allows us to represent time using a range from 0 to 11, corresponding to 12-hour intervals on the clock face. Any time beyond 11:59 PM (23:59) wraps around and starts at 0:00 (midnight) again. In this specific example, by convention, we use the AM/PM suffix to differentiate between the first zero hour (0=12 am) and the second zero hour (0=12 pm).

By using modular arithmetic with modulo 12, we simplify the representation of time on a 24-hour clock into a more familiar 12-hour format, helping us understand and communicate time more easily.

JB

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We can create a CYCLIC GROUP of NINE, using the set of numbers from 0 to 8, with the operation being addition modulo 9. This group satisfies all four group axioms:

1. Closure: The sum of any two numbers in the set, when reduced modulo 9, will result in another number in the set.
2. Associativity: For any three numbers a, b, and c in the set, the equation (a + b) + c = a + (b + c) holds.
3. Identity element: The number 0 acts as the identity because adding 0 to any number in the set doesn’t change the number.
4. Inverse element: For each number ‘a’ in the set, there is another number ‘b’ in the set such that a + b (modulo 9) = 0. For instance, the inverse of 1 is 8 because 1 + 8 = 9, which is equivalent to 0 modulo 9.

This group is known as a cyclic group. The SERVANT LEADER of the group is the ZERO.

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Eight complementary pairs out of the 16 Keirsey types

It would be reasonable to say that Artisan and Guardian temperaments, as well as Rationals and Idealists, are primary complementary polarities. This is based on the characteristics and behavioral patterns associated with each temperament. This framework of understanding human behavior was developed by David Keirsey, who divided these four temperaments into 16 personality types, providing a more detailed and nuanced understanding of individual personalities.

Creating eight complementary pairs from the 16 Keirsey types requires considering the unique characteristics of each type. These pairs are typically formed by matching an introverted type with an extroverted type from the same temperament group. Here’s a possible arrangement:

1. Artisans (SP):
   • ISTP (Crafter) and ESTP (Promoter)
   • ISFP (Composer) and ESFP (Performer)
2. Guardians (SJ):
   • ISTJ (Inspector) and ESTJ (Supervisor)
   • ISFJ (Protector) and ESFJ (Provider)
3. Idealists (NF):
   • INFP (Healer) and ENFP (Champion)
   • INFJ (Counselor) and ENFJ (Teacher)
4. Rationals (NT):
   • INTP (Architect) and ENTP (Inventor)
   • INTJ (Mastermind) and ENTJ (Fieldmarshal)

This way, each pair has similar values and approaches to life (due to being from the same temperament group), but their introversion/extroversion difference can provide a balance, thereby making them complementary. Likewise, within each temperament, we could also pair complementary polarities for each introversion and extroversion type, according to the three secondary axes: S or N, F or T, and P or J, respectively.

The idea of complementarity in personality types is subjective and varies greatly depending on the context (work, relationships, etc.). In some contexts, similarity may be more beneficial than complementarity.

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On the fifth Ray of Concrete Knowledge or Science, we find the Master Hilarion, who, in an earlier incarnation was Paul of Tarsus. … His is the energy which, through His disciples, is stimulating the Psychical Research groups everywhere, and He it was Who initiated, through various pupils of His, the Spiritualistic movement. He has under observation all those who are psychics of the higher order, and assists in developing their powers for the good of the group, and in connection with certain of the devas of the astral plane He works to open up to the seekers after truth that subjective world which lies behind the grossly material.

Initiation, Human and Solar by Alice A. Bailey, p. 59

I have said earlier that the intersecting energies in the etheric body of the planet are at this time a network of squares. When the creative process is complete and evolution has done its work, these squares will become a network of triangles. Necessarily this is a symbolic way of speaking. In the Book of Revelations which was dictated 1900 years ago by the disciple who is now known as the Master Hilarion, reference is made to the “city which stands four-square.” The etheric vehicle of the planet was inherited from a former solar system, with the purpose or intention in view of its transformation into a network of triangles in this solar system. In the next one of the triplicity of solar systems (the third or last) in which the will of God works out, the etheric body will begin as a network of triangles, but this will be resolved into a network of interlinked circles or of linked rings, indicating the fulfilment of interlocking relationships. In this present system, the result of evolution, as far as the etheric body is concerned, will be the contact established between all three points of each triangle, making a ninefold contact and a ninefold flow of energy; this is consistent with the fact that nine is the number of initiation, and by the time the destined number of disciples have taken the nine possible initiations, this triangular formation of the planetary etheric body will be complete.

The idea can be conveyed symbolically by the diagram below, which pictures the triangular formation and the mode of a dual growth or progression and expansion of the network because, starting with the initial triangle, two points only are left for the processes of extension. [Page 164]

Telepathy and the Etheric Vehicle by Alice A. Bailey

The initial triangle was formed by Sanat Kumara, and we call the three energies which circulate through its medium the three major Rays of Aspect. The four Rays of Attribute formed their own triangles and yet, in a paradoxical manner, they are responsible for the “squares” through which their energies at this time pass. Thus the work of transforming the inherited etheric body was begun and has proceeded ever since. In the etheric body of the human being you have a repetition of the same process in the triangle of energies created by the relationship between the Monad, the Soul, and the Personality.

Ibid.

Two interlaced tetrahedra form the eight vertices of a cube (hexahedron).

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Recapitulating:

Each Unit of Service within the New Group of World Servers is a GROUP with an inner mathematical and fractal structure reproducing the planned outer planetary etheric body for this cycle: the TRIANGLE transformed from a “cuboctahedron.” At the heart of each Unit is the Zero, the servant leader. Groups of nine awakened aspirants (the zero plus eight) can then be occultly absorbed in the heart center of the planetary Logos.

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I am the ONE,
Begotten by the ZERO,
Differentiated into INFINITY,
Integrated into the conscious SYNTHESIS
Of LIGHT, and LOVE and LIFE,
For the GLORY of the ONE.

https://www.ngsm.org/agni-yoga/IHF-TP-1.htm

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POSTSCRIPTUM

The Father-Mother Duad is exponentiated by three (2^3=8) to create the Odgoad.

The Ogdoad consists of four duads, representing different aspects of the primeval existence.

1. Nun and Naunet – Representing the primordial waters.
2. Heh and Hauhet – Symbolizing infinity or eternity.
3. Kek and Kauket – Representing darkness.
4. Amun and Amaunet – Signifying hiddenness or obscurity.

The world was said to have emerged from the interaction of these four dual elements, specifically from a lotus flower that bloomed on the primordial mound that rose from the watery chaos (symbolized by Nun and Naunet).

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The Magic of Number Nine
(in the “perfect” base-10 number system)

The number nine possesses an intriguing property in mathematics. When multiplied by any other number, the digits of the resulting product always sum to nine or a multiple of nine. This characteristic lends a certain cyclicality to the number, echoing a sense of regeneration.

For example, if we take 9 and multiply it by 5, we get 45. The sum of the digits 4 and 5 equals 9. Similarly, if we multiply 9 by 12, the result is 108, and adding 1, 0, and 8 yields 9 again.

This pattern is unbroken, consistent, and dependable, much like the cycle of regeneration itself. It’s this underlying consistency that has led to nine being referred to as the number of regeneration.

JB

Again the numerical significance of the numbers must be studied; this time they are hid in the number ten, or three cycles of three lesser periods, each making nine, and one synthezising period, leading up to the consummation of one period within the greater cycle; this is signified by the ten of relative perfection.

The number of the deva evolution is six, as that of man is now five, and as ten stands for perfected man, so twelve stands for perfection in the deva kingdom.

A Treatise on Cosmic Fire by Alice A. Bailey

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