Mathematical Descriptions of Models A and G

The Transitive Permutation Group $8T_9$

The transitive permutation group $8T_9$ is the 9th transitive subgroup of the symmetric group $S_8$ (acting on 8 points) in the database of transitive groups. It is known to be isomorphic to the direct product of the dihedral group of order 8 and the cyclic group of order 2 ($D_4\times {\mathbb{Z}}_2$).

Model A

In Model A, we have a set $F_{MA}$ of 8 functions (labelled points) and a set

$I=\{\mathrm{Ne},\mathrm{Si},\mathrm{Fe},\mathrm{Ti},\mathrm{Se},\mathrm{Ni},\mathrm{Te},\mathrm{Fi}\}$

of 8 information metabolism elements (IMEs) occupying $F_{MA}$. The group which comprises the classical intertype relations $\mathbb{S}\cong D_4\times {\mathbb{Z}}_2$ acts on functions transitively by permuting them in specific ways depending on the action applied (i.e. the relationship that acts on Model A by permuting its set of functions). For instance, the kindred relationship $k\in \mathbb{S}$ exchanges functions 2 and 4, but also functions 6 and 8. Therefore, we can denote this as $(2,4)(6,8)$. An order 4 relationship like the supervisee $s\in \mathbb{S}$ shifts functions 1-4 as well as functions 5-8, cyclically, which can be expressed as $(1,2,3,4)(5,6,7,8)$.

$\mathbb{S}$ can be described as the group generated by certain permutations such as $s=(1,2,3,4)(5,6,7,8)$, $d=(1,5)(2,6)(3,7)(4,8)$, and $a=(1,6)(2,5)(3,8)(4,7)$. That is, $s(1)=2$, $s(2)=3$, etc. From these generators, it follows directly that this permutation group is isomorphic to $D_4\times {\mathbb{Z}}_2$, that acts on 8 points, and hence Model A realises the transitive permutation group $8T_9$.

The Transitive Permutation Group $8T_2$

The transitive permutation group $8T_2$ is the 2nd transitive subgroup of the symmetric group $S_8$ (acting on 8 points) in the database of transitive groups. It is known to be isomorphic to the direct product of the cyclic groups of orders 2 and 4 (${\mathbb{Z}}_2\times {\mathbb{Z}}_4$, respectively).

Model G

In Model G, we have a set $F_{MG}$ of 8 functions (labelled points) and a set

$\mathcal{E}=\{?\mathrm{N}{\mathrm{e}}^{+},!\mathrm{S}{\mathrm{i}}^{-},!\mathrm{F}{\mathrm{e}}^{+},?\mathrm{T}{\mathrm{i}}^{-},?\mathrm{F}{\mathrm{e}}^{-},!\mathrm{T}{\mathrm{i}}^{+},!\mathrm{S}{\mathrm{e}}^{-},?\mathrm{N}{\mathrm{i}}^{+},?\mathrm{S}{\mathrm{e}}^{+},!\mathrm{N}{\mathrm{i}}^{-},!\mathrm{T}{\mathrm{e}}^{+},?\mathrm{F}{\mathrm{i}}^{-},?\mathrm{T}{\mathrm{e}}^{-},!\mathrm{F}{\mathrm{i}}^{+},!\mathrm{N}{\mathrm{e}}^{-},?\mathrm{S}{\mathrm{i}}^{+}\}$

of 16 monadic (signed) elements, half of which occupy $F_{MG}$, depending on whether a type is Process or Result. The set of functions is acted on by the index-2 subgroup of $\mathbb{S}$ that describes Process/Result, for which we’ll denote $H$, and is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$. $H$ acts on functions in specific ways depending on the action applied, for a particular type $t\in T$.

Like with $\mathbb{S}$, $H$ can be described by certain permutations such as $B=(1,2,3,4)(5,6,7,8)$ for order 4 relationships and $d=(1,6)(2,7)(3,8)(4,5)$ for order 2 relationships. From these generators, it follows directly that this permutation group is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$, that acts on 8 points, and hence Model G realises the transitive permutation group $8T_2$, when restricted to a single orbit.

Since Model-G does not model all elements of the full socion for a type $t\in T$, the construction of Model G layouts for both process and result types, for which we’ll denote $P$ and $R$, respectively, are needed to represent all information in the Socion. Thus, Model-G can be formally understood as $F_{MG}\times \{P,R\}$ with $H$ acting on $F_{MG}$, the selector dichotomy (Process/Result) acting on $\{P,R\}$ and no group action mixing the two.

Alternate Interpretations of Model A (Systems of Injections)

The set $I$ is sometimes substituted for the set $\mathcal{E}$, and consequently the acting group on Model A is no longer $\mathbb{S}$, but an index-2 subgroup of it. For example, in the School of Classical Socionics (SCS), the set of functions is acted on by $H\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (Process/Result), so in this sense Models A and G are isomorphic (both $8T_2$ groups). However, in the School of System Socionics (SSS), the set of functions is acted on by the Asking/Declaring subgroup, which is isomorphic to $D_4$ (the dihedral group of order 8). In this sense, Model A can be can be decomposed into the two orbits of $\mathbb{S}$, one of which can be described as the transitive permutation group $8T_4$, which is a subgroup of $S_8$ and is isomorphic to $D_4$.

Finally, the ‘Presence Cube’ model demonstrates the constructed vector space on Model A. $F_{MA}$ has size $2^3$ so we can construct a dichotomy system on it. There are only two order 8 subgroups of $\mathbb{S}$, which are isomorphic to the vector space ${\mathbb{Z}}_2^3$, namely Democratic/Aristocratic and Irrational/Rational, denoted $D$ and $J$. Of the two, only $D$ acts transitively on $F_{MA}$ (e.g. Ne can occur in any position in a Democratic type), and the actions of $D$ on $F_{MA}$ are faithful to $\mathbb{S}$. In this sense, this construction of Model A can be can be decomposed into the two orbits of $\mathbb{S}$, one of which can be described as the transitive permutation group $8T_3$, which is a subgroup of $S_8$ and is isomorphic to ${\mathbb{Z}}_2^3$.

Technical note: Systems of injections (charge models with only 8 functions) are best viewed as orbit models. Any compatible index-2 subgroup $H$ acts transitively on a single orbit of size 8 (hence an $8T_k$ label applies). The full charge construction $F\times \{A,B\}$ is an intransitive $H$-set with two orbits; it is not itself an $n{T}_k$ object unless extended by a sheet-swapping element, yielding a transitive $16T_l$ action.

What counts as an acceptable system of injections?

An acceptable system of injections on Model A must arise from an index-2 subgroup of $\mathbb{S}$ corresponding to a supralocal dichotomy. These include Democratic/Aristocratic, Asking/Declaring, Process/Result, and Positivist/Negativist.

The reason for this restriction is structural. Each information element $i\in I$ admits a decomposition into its associated monadic elements in $\mathcal{E}$, and any admissible injection must preserve this correspondence. Equivalently, for every function in $F_{MA}$, either the unsplit element $i$ or exactly one of its associated monadic refinements must be present, uniformly across all types.

Index-2 subgroups not associated with supralocal dichotomies (these are the general dichotomies, which when intersected define the ‘Temperament’ tetrachotomy) fail to satisfy this condition, as they induce splittings that cannot be reconciled with the monadic decomposition of elements in $I$. Consequently, no other index-2 subgroup of $\mathbb{S}$ satisfies the uniform monadic-refinement constraint above.

Function Dichotomies

Classical

Model A

$\begin{array}{cccc}\text{Intertype Relations}& \text{Function Dichotomy}& \text{1st set}& \text{2nd set}\\ \{\{e,k\},\{m,s\},\{g,l\},\{c,S\}\}& \text{Mental/Vital}& \{1,2,3,4\}& \{5,6,7,8\}\\ \{\{e,k\},\{g,l\},\{d,h\},\{x,i\}\}& \text{Accepting/Producing}& \{1,3,5,7\}& \{2,4,6,8\}\\ \{\{e,k\},\{g,l\},\{a,b\},\{q,B\}\}& \text{Bold/Cautious}& \{1,3,6,8\}& \{2,4,5,7\}\\ \{\{e,k\},\{m,s\},\{x,i\},\{q,B\}\}& \text{Strong/Weak}& \{1,2,7,8\}& \{3,4,5,6\}\\ \{\{e,k\},\{m,s\},\{d,h\},\{a,b\}\}& \text{Valued/Subdued}& \{1,2,5,6\}& \{3,4,7,8\}\\ \{\{e,k\},\{c,S\},\{a,b\},\{x,i\}\}& \text{Inert/Contact}& \{1,4,6,7\}& \{2,3,5,8\}\\ \{\{e,k\},\{c,S\},\{d,h\},\{q,B\}\}& \text{Evaluatory/Situational}& \{1,4,5,8\}& \{2,3,6,7\}\end{array}$

Model G

$\begin{array}{cccc}\text{Intertype Relations}& \text{Function Dichotomy}& \text{1st set}& \text{2nd set}\\ \{e,B,g,b\}& \text{External/Internal}& \{1,2,3,4\}& \{5,6,7,8\}\\ \{e,g,d,x\}& \text{Stable/Unstable}& \{1,3,6,8\}& \{2,4,5,7\}\\ \{e,g,s,S\}& \text{Opening/Closing}& \{1,3,5,7\}& \{2,4,6,8\}\\ \{e,B,m,x\}& \text{Leading/Following}& \{1,2,5,8\}& \{3,4,6,7\}\\ \{e,b,m,d\}& \text{Accelerating/Decelerating}& \{1,4,5,6\}& \{2,3,7,8\}\\ \{e,b,S,x\}& \text{Values/Tools}& \{1,4,7,8\}& \{2,3,5,6\}\\ \{e,B,d,s\}& \text{Automatic/Conscious}& \{1,2,6,7\}& \{3,4,5,8\}\end{array}$

Gulenko-Newman

Model A

$\begin{array}{cccc}\text{Intertype Relations}& \text{Function Dichotomy}& \text{1st set}& \text{2nd set}\\ \{\{e,k\},\{m,s\},\{g,l\},\{c,S\}\}& \text{Mental/Vital}& \{1,2,3,4\}& \{5,6,7,8\}\\ \{\{e,k\},\{g,l\},\{d,h\},\{x,i\}\}& \text{Accepting/Producing}& \{1,3,5,7\}& \{2,4,6,8\}\\ \{\{e,k\},\{g,l\},\{a,b\},\{q,B\}\}& \text{Bold/Cautious}& \{1,3,6,8\}& \{2,4,5,7\}\\ \{\{e,k\},\{m,s\},\{d,h\},\{q,B\}\}& \text{Energetic/Informational}& \{1,2,5,8\}& \{3,4,6,7\}\\ \{\{e,k\},\{c,S\},\{d,h\},\{a,b\}\}& \text{Impressionable/Unimpressionable}& \{1,4,5,6\}& \{2,3,7,8\}\\ \{\{e,k\},\{c,S\},\{x,i\},\{q,B\}\}& \text{Tensioned/Relaxed}& \{1,4,7,8\}& \{2,3,5,6\}\\ \{\{e,k\},\{m,s\},\{a,b\},\{x,i\}\}& \text{Excitable/Inhibitable}& \{1,2,6,7\}& \{3,4,5,8\}\end{array}$

Model G

$\begin{array}{cccc}\text{Intertype Relations}& \text{Function Dichotomy}& \text{1st set}& \text{2nd set}\\ \{e,B,g,b\}& \text{External/Internal}& \{1,2,3,4\}& \{5,6,7,8\}\\ \{e,g,d,x\}& \text{Stable/Unstable}& \{1,3,6,8\}& \{2,4,5,7\}\\ \{e,g,s,S\}& \text{Opening/Closing}& \{1,3,5,7\}& \{2,4,6,8\}\\ \{e,B,m,d\}& \text{Energetic/Informational}& \{1,2,5,6\}& \{3,4,7,8\}\\ \{e,b,d,S\}& \text{Impressionable/Unimpressionable}& \{1,4,6,7\}& \{2,3,5,8\}\\ \{e,B,S,x\}& \text{Tensioned/Relaxed}& \{1,2,7,8\}& \{3,4,5,6\}\\ \{e,b,s,x\}& \text{Excitable/Inhibitable}& \{1,4,5,8\}& \{2,3,6,7\}\end{array}$

The Conceptual Contrast between Model A and Model G

Model A is a surjective coset model: functions correspond to cosets of the ‘kindreds’ subgroup $\{e,k\}$ of $\mathbb{S}$, hence 8 IMEs. However, Model G is an injective refinement model: functions receive monadic elements via selector-controlled injections.

The Orbit-Stabiliser Theorem

Let $G$ be a group which acts on a finite set $X$.
Let $x\in X$.
Let $\mathrm{Orb}(x)$ denote the orbit of $x$.
Let $\mathrm{Stab}(x)$ denote the stabiliser of $x$ by $G$.
Let $[G:\mathrm{Stab}(\mathrm{x})]$ denote the index of $\mathrm{Stab}(x)$ in $G$.

Then:

$|\mathrm{Orb}(x)|=[G:\mathrm{Stab}(x)]=\frac{|G|}{|\mathrm{Stab}(x)|}$

Systems of Injections

Let $G:=\mathbb{S}$ act intransitively on a finite set $X$ containing 16 points with two different orbits, which are the cosets of an index-2 subgroup $H\le \mathbb{S}$, and thus the orbit size is $|\mathrm{Orb}(x)|=8$. A set of mapped functions is denoted $F\times \{A,B\}$ where $A$ and $B$ are the cosets of $H$. Since an information element $i\in I$ decomposes into two monadic elements, which are cosets of the kindred subgroup $\{e,k\}$, the stabiliser subgroup when $x$ is the Base Function of Model A will also be the subgroup $\{e,k\}$, and this applies to all accepting functions as applying the kindred operator $k$ would fix the functions 1, 3, 5, and 7, and applying the lookalike operator $l$ would fix functions 2, 4, 6 and 8, and only changing the charge of the element in systems of injections. So, for producing functions, the stabiliser subgroup is $\{e,l\}$. $|\mathrm{Stab}(x)|=2$ because $\frac{|G|}{|\mathrm{Orb}(x)|}=\frac{16}{8}=2$. So for odd positions, the stabiliser is conjugate to $\{e,k\}$, whereas for even positions, the stabiliser is conjugate to $\{e,l\}$.

References / Further Reading

• https://sedecology.com/math
• A Representation-Theoretic Framework for Intertype Relations in Socionics
• https://people.maths.bris.ac.uk/~matyd/GroupNames/T31.html
• https://sedecology.blogspot.com/2020/07/the-presence-cube.html
• https://varlawend.blogspot.com/2023/06/encyclopedia-of-model-g-and-model.html

See Also

• Injections - Model A and Model G Charge Systems
• Injections - Function Dichotomies to TIM Tetrachotomies
• Element (Information) Dichotomies
• https://proofwiki.org/wiki/Orbit-Stabilizer_Theorem