Let \(k\) be a commutative ring and \(G\) a group. The Group Algebra \(k[G]\) is the free \(k\)-module with basis \([g]\) indexed by \(g\in G\), equipped with \[ [g][h]=[gh],\qquad 1=[1_G],\qquad (\sum _g a_g[g])(\sum _h b_h[h])=\sum _{g,h}a_gb_h[gh]. \] The scalar embedding sends \(a\) to \(a[1_G]\). Thus each group element is a unit of \(k[G]\), with inverse \([g^{-1}]\).

If \(A\) is a \(k\)-algebra and \(u:G\to A^\times \) is a group homomorphism, there is a unique \(k\)-algebra homomorphism \[ \widetilde u:k[G]\longrightarrow A, \qquad \widetilde u\left (\sum _g a_g[g]\right )=\sum _g a_g u(g). \] This universal property extends representations and concrete group maps from basis elements to the whole Group Algebra. See [sengupta2010representations, secs. 3.1--3.2, pp. 39--41].

Let \(k\) be a field, \(G\) a group, and \(V\) a \(k\)-vector space. Giving a linear representation \(\rho :G\to \operatorname {GL}_k(V)\) is equivalent to giving a unital left \(k[G]\)-module structure on \(V\), up to the identity type synonyms used to keep the 2 scalar actions distinct. The action associated to \(\rho \) is \[ (\sum _g a_g[g])\cdot v=\sum _g a_g\rho (g)v. \] Conversely, restriction of a unital \(k[G]\)-action to the units \([g]\) recovers \(\rho \). Invariant subspaces are submodules and intertwining linear maps are \(k[G]\)-linear maps.

Let \(G\) act on \(k[G]\) by left multiplication. In the basis \(\{[h]:h\in G\}\), an element \(g\) sends \([h]\) to \([gh]\); hence its matrix is the permutation matrix of left translation. This is the left regular representation. It is faithful because the image of \([1_G]\) records \(g\).

The same basis underlies the coefficient calculations in the real Group Algebra of the quaternion group: there, the quaternion coordinate recovers antisymmetric coefficient pairs and the character coordinates recover symmetric pairs. The regular module is the linear carrier for both the abstract Group Algebra and the explicit 8-coordinate computation. Compare [james2001representations, ch. 6, pp. 53--58].

Let \(G\) be finite and \(k\) a field in which \(|G|\) is invertible. Every subrepresentation \(W\subseteq V\) has a \(G\)-stable complement. Consequently every finite-dimensional \(k[G]\)-module, and \(k[G]\) itself, is semisimple.

Let \(G\) be finite and \(k\) an algebraically closed field whose characteristic does not divide \(|G|\). There are positive integers \(d_C\), indexed by the conjugacy classes \(C\) of \(G\), such that \[ k[G]\cong _{k\text {-alg}} \prod _{C\in \operatorname {Conj}(G)}M_{d_C}(k), \qquad \sum _C d_C^2=|G|. \] The class labels are an indexing choice: the statement does not canonically pair a particular conjugacy class with a particular irreducible module.

Maschke semisimplicity followed by Artin--Wedderburn gives the stated product. Lux and Pahlings state the results in [lux2010representations, thms. 1.5.5--1.5.6, pp. 56--57].

The dimension identity can also be read from the regular representation, where an irreducible module occurs with multiplicity equal to its dimension; compare [james2001representations, thms. 11.9 and 11.12, pp. 100--101].

For a finite-dimensional representation \(\rho \) over a characteristic-zero field, its character is \(\chi _\rho (g)=\operatorname {tr}(\rho (g))\). Trace is unchanged by conjugation, so \(\chi _\rho \) is a class function. A character table places the irreducible characters in rows and the conjugacy classes in columns. Tau Ceti identifies class functions with functions on conjugacy classes in TauCeti.ClassFunction.equivConjClasses.

Over \(\mathbb C\), the irreducible characters form an orthonormal basis of the class functions. Thus the table is square: the number of irreducible characters equals the number of conjugacy classes. This also counts the Wedderburn blocks and gives the dimension of the center. See [james2001representations, thm. 16.4, pp. 159--166].

Over a splitting field of characteristic not dividing \(|G|\), an irreducible character \(\chi \) defines the central element \[ e_\chi =\frac {\chi (1)}{|G|}\sum _{g\in G}\chi (g^{-1})[g]. \] It is a nonzero primitive central idempotent. On a simple module with character \(\psi \), it acts as the identity if \(\psi =\chi \) and as zero otherwise. Distinct \(e_\chi \) are therefore orthogonal block projectors. See [lux2010representations, thm. 2.1.6 and cor. 2.1.7, pp. 88--90]. A constructed family of such projectors does not yield a complete block decomposition until a sum-to-1 statement is also known.

Let \(\rho \) be an irreducible finite-dimensional representation of a finite group over an algebraically closed field of characteristic 0. Its Frobenius--Schur indicator \[ \nu _2(\rho )=\frac 1{|G|}\sum _{g\in G}\chi _\rho (g^2) \] takes exactly 1 of the values \(1\), \(0\), and \(-1\). The value \(1\) is equivalent to the existence of a nondegenerate invariant symmetric bilinear form; \(-1\) is equivalent to a nondegenerate invariant alternating form; and \(0\) is equivalent to the absence of a nonzero invariant bilinear form. See [james2001representations, def. 23.13, the proof of thm. 23.14, and thm. 23.16, pp. 273--277]. James and Liebeck state the alternatives using nonzero invariant forms; for an irreducible representation, a nonzero invariant symmetric or alternating form has zero radical and is therefore nondegenerate.

This trichotomy classifies the invariant-form behavior of the complex irreducible representation. It does not by itself compute a Schur index or a complete decomposition after descent to \(\mathbb R\).

Over \(\mathbb R\), a simple finite-dimensional block may instead have division algebra \(\mathbb R\), \(\mathbb C\), or \(\mathbb H\). The Frobenius--Schur trichotomy in the Frobenius--Schur trichotomy detects invariant-form type, but it is not a complete theorem about a character's field of values or Schur index. The explicit decomposition \[ \mathbb R[Q_8]\cong \mathbb R^4\times \mathbb H \] in the real Group Algebra of the quaternion group is a complete real example, not an application of the algebraically closed formula. For Schur indices and invariant forms, see [lux2010representations, sec. 2.9, pp. 164--172].

Let \(N\trianglelefteq G\), let \(V\) be a \(G\)-representation, and let \(W\subseteq V|_N\) be an irreducible \(N\)-subrepresentation. For \(g\in G\), the conjugate representation \(,{}^gW\) has the same vector space and action \[ n\cdot _g w=(g^{-1}ng)\cdot w. \] The inertia subgroup of \(W\) is \[ I_G(W)=\{g\in G:{}^gW\cong W\}. \] The isomorphism classes of conjugates form a \(G\)-orbit, with stabilizer \(I_G(W)\). See [lux2010representations, sec. 3.6, pp. 222--225].

Let \(N\trianglelefteq G\) be finite, and work over a field for which the relevant finite-group modules are semisimple. If \(V\) is an irreducible \(G\)-representation, then the irreducible constituents of \(V|_N\) lie in 1 \(G\)-orbit under conjugation. Equivalently, after choosing 1 constituent \(W\), every other constituent is isomorphic to some \(,{}^gW\).

This is the orbit assertion in Clifford theory. It does not by itself say that all constituents occur with the same multiplicity, nor does it construct extensions across the inertia subgroup. Compare [lux2010representations, thm. 3.6.2(a), pp. 223--225].

Let \(N\trianglelefteq G\) and let \(V\) be an irreducible complex representation of \(G\). If \(W_1,\ldots ,W_r\) represent the distinct conjugates of an irreducible constituent of \(V|_N\), then for 1 positive integer \(e\), \[ V|_N\cong e(W_1\oplus \cdots \oplus W_r). \] Thus restriction is homogeneous along a single orbit. The common multiplicity is an additional assertion beyond the single orbit in Clifford restriction. This is Clifford's theorem, [james2001representations, thm. 20.8, pp. 216--217].

The inertia subgroup is the intermediate group where one studies whether \(W\) extends, possibly projectively, before inducing to \(G\). For semidirect products, this reduces the problem to the action of the complement. For \(Q_8\rtimes C_3\cong 2T\), it connects the normal quaternion subgroup to the skew-Group-Algebra model in two routes from binary tetrahedral symmetry. This theorem does not supply the full character table of \(2T\).

Suppose \(G=N\rtimes H\). Conjugation by \(H\) permutes the irreducible constituents of \(V|_N\). Their stabilizers are the inertia subgroups, so the orbit and homogeneity statements in the single orbit in Clifford restriction and Clifford homogeneity and the semidirect route reduce the search for irreducible \(G\)-representations to stabilizer data, projective extension, and induction. Clifford theory thereby gives a character-theoretic route for semidirect products.

For \(Q_8\rtimes C_3\cong 2T\), the action and ordinary skew-Group-Algebra realization are recorded in the central split of the real Group Algebra of 2T and two routes from binary tetrahedral symmetry. They supply the normal-subgroup and algebraic input to this route, but do not give the full character table of \(2T\).

A finite presentation \(\langle x_1,\ldots ,x_m\mid r_1,\ldots ,r_s\rangle \) gives a compact input for algorithms that enumerate cosets, compute conjugacy classes, and build character data. Sims develops the algorithms and the conditions under which such computations terminate; Lux and Pahlings place them inside computational representation theory.

The output has 3 possible evidential strengths. A transcript with versioned inputs is reproducible. A compact certificate, such as matrices satisfying the relations together with independently checked completeness identities, can be verified without trusting the search. An unrecorded software answer is neither. GAP is therefore a discovery and calculation tool; its output becomes mathematics here only when the decisive relations and completeness checks are visible. See [sims1994computation, ch. 1] and [lux2010representations, secs. 1.1 and 4.2].

The route is \[ \begin {aligned} \text {finite presentation}&\longrightarrow \text {computed candidates}\\ &\longrightarrow \text {checked relations and completeness}. \end {aligned} \] The arrows distinguish generation from certification rather than describing a project workflow.

Let \(k\) be a field, \(G\) a group, and \(\alpha :G\times G\to k^\times \) a normalized 2-cocycle: \[ \alpha (1,g)=\alpha (g,1)=1, \qquad \alpha (g,h)\alpha (gh,\ell )= \alpha (h,\ell )\alpha (g,h\ell ). \] The twisted Group Algebra \(k_\alpha [G]\) has basis \(u_g\) and multiplication \[ u_gu_h=\alpha (g,h)u_{gh}. \] The cocycle equation is exactly the associativity condition.

A projective representation with factor set \(\alpha \), namely operators \(T_g\) satisfying \(T_gT_h=\alpha (g,h)T_{gh}\), extends uniquely to an algebra map \(k_\alpha [G]\to \operatorname {End}_k(V)\); conversely an algebra map gives such a projective representation on its basis operators. Cohomologous cocycles rescale the basis and give isomorphic twisted algebras. See [conlon1964twisted, the introduction and sec. 1, pp. 152--155].

A generalized Clifford algebra becomes a twisted group algebra only after its parameters are fixed. Let \(k\) contain a primitive \(n\)-th root of unity \(\omega \), assume \(\operatorname {char}k\nmid n\), and choose \(q_1,\ldots ,q_m\in k^\times \). The algebra generated by \(e_1,\ldots ,e_m\) with \[ e_i^n=q_i, \qquad e_i e_j=\omega e_j e_i\quad (j<i) \] is isomorphic to a twisted group algebra of \((\mathbb Z/n\mathbb Z)^m\) for an explicit 2-cocycle. The monomials in the \(e_i\) correspond to its twisted basis. See [cheng2019new, sec. 2.3 and prop. 2.1, pp. 3--4].

This is a qualified bridge, not an identification of every Clifford algebra with an ordinary Group Algebra. The field, root of unity, grading, parameters, and twisting are part of the theorem. It is also different from Clifford theory in the single orbit in Clifford restriction and Clifford homogeneity and the semidirect route, which concerns restriction of representations to normal subgroups.

For a constituent \(W\) with inertia subgroup \(I=I_G(W)\), choose intertwiners \(T_i:W\to {}^iW\). Their composites need not satisfy the group law strictly. Instead one obtains scalars \[ T_iT_j=\alpha (i,j)T_{ij}. \] Associativity makes \(\alpha \) a factor set. Changing the intertwiners changes \(\alpha \) by a coboundary. The resulting cohomology class measures the obstruction to replacing the projective action by an ordinary action.

Twisted Group Algebras therefore enter the Clifford-theory route. The ordinary skew Group Algebra in two routes from binary tetrahedral symmetry corresponds to trivial twisting; it should not be identified with a general \(k_\alpha [G]\). The extension and induction analysis is developed in [lux2010representations, secs. 3.6--3.7, pp. 222--240].

For each \(g\in G\), let \(A_g=k[g]\) be the 1-dimensional subspace spanned by the basis element \([g]\). Then \[ k[G]=\bigoplus _{g\in G}A_g, \qquad A_gA_h\subseteq A_{gh}. \] Thus \(k[G]\) is canonically graded by the group \(G\). In Lean's additive graded-algebra interface this statement is expressed after passing from the multiplicative index \(G\) to its additive copy; each single basis element lies in its corresponding grade.

The basis and multiplication law in the Group Algebra and its universal extension give the decomposition and product containment directly; compare the construction in [sengupta2010representations, secs. 3.1--3.2, pp. 39--41]. It is distinct from the Hopf structure in the standard Hopf structure on a Group Algebra: the grading records where products land, while the coalgebra maps record how a basis element is copied and inverted.

Let \(k\) be a commutative ring and \(G\) a group. The Group Algebra \(k[G]\) is a Hopf algebra with structure determined on basis elements by \[ \Delta ([g])=[g]\otimes [g],\qquad \epsilon ([g])=1,\qquad S([g])=[g^{-1}]. \] The comultiplication and counit are algebra homomorphisms; the antipode extends inversion and satisfies the 2 convolution identities.

The general route has several distinct endpoints. Over an algebraically closed field, Wedderburn blocks and the character table organize all simple representations. Over \(\mathbb R\), division-algebra type and descent data enter. For a normal subgroup, Clifford theory organizes restriction and induction. With a nontrivial factor set, projective representations are modules over a twisted algebra.

The quaternion calculation in the real Group Algebra of the quaternion group closes 1 bounded endpoint: it gives an explicit real-algebra equivalence \(\mathbb R[Q_8]\cong \mathbb R^4\times \mathbb H\), and Real blocks of the quaternion Group Algebra identifies its center. It does not prove a general real Wedderburn theorem, the characters of \(2T\), or the Clifford-theory extension step. Those are separate statements and do not follow from the worked example.

Wilson's proposed route begins with \[ Q_8\rtimes C_3\cong 2T \] and then studies the iterated semidirect product \((G_{27}\rtimes Q_8)\rtimes C_3\), with the indicated actions part of the data. Its central scalar subgroup \(C_3\leq G_{27}\) has quotient the Hessian group of order 216, so the larger group is a triple cover of that quotient. The name alone would not choose either semidirect action. See [wilson2024discrete, sec. 2.2, pp. 4--5].

The comparison with \(U(1)\times SU(2)\times SU(3)\) is an attributed finite-model proposal, not an isomorphism or a canonical physical interpretation. To become a mathematical correspondence it would need typed representations and homomorphisms, the preserved forms or observables, and a rule connecting them to physical data. Hamilton's geometric-algebra models in [hamilton2023supergeometric] and [hamilton2023unification] provide a comparison motivation, not independent confirmation of this finite-group construction.

Let \(C_3=\langle g\mid g^3=1\rangle \). Sending \(g\) to \(x\) gives \[ \mathbb {R}[C_3]\cong \mathbb {R}[x]/(x^3-1). \] Over the real numbers, \[ x^3-1=(x-1)(x^2+x+1), \] and the two factors are coprime. The Chinese remainder theorem and the identification \(\mathbb {R}[x]/(x^2+x+1)\cong \mathbb {C}\) therefore give \[ \mathbb {R}[C_3]\cong \mathbb {R}\times \mathbb {C}. \]

More concretely, if \(\zeta \) is a primitive complex cube root of unity, the isomorphism is \[ a+bg+cg^2\longmapsto \bigl (a+b+c,\ a+b\zeta +c\zeta ^2\bigr ). \] The two factors can be read in three compatible ways: \[ \begin {array}{c|c|c} \text {factor of }x^3-1& \text {real algebra block}& \text {real representation}\\ \hline x-1&\mathbb {R}&\text {trivial line}\\ x^2+x+1&\mathbb {C}&\text {rotation plane}. \end {array} \] The rotation plane is the linearization of the nontrivial part of the regular Group Action. More explicitly, the generator cyclically permutes the basis \(1,g,g^2\). The averaging element \[ e_0=\frac {1+g+g^2}{3} \] projects onto the fixed line \(\mathbb {R}(1+g+g^2)\), while its complementary kernel is the augmentation plane \[ \{a+bg+cg^2:a+b+c=0\}. \] Orbits, freeness, and transitivity belong to the action; the projector and invariant subspaces appear only after linearization.

Since the spectrum of a product is the disjoint union of the spectra, \[ \operatorname {Spec}\mathbb {R}[C_3] \cong \operatorname {Spec}\mathbb {R} \sqcup \operatorname {Spec}\mathbb {C}. \] This is a useful commutative picture. It is not yet a reason to introduce a topos, and it does not extend without choices to noncommutative Group Algebras.

The quaternion group \(Q_8\) has a real Group Algebra decomposition \[ \mathbb {R}[Q_8]\cong \mathbb {R}^4\times \mathbb {H}. \] This can be seen directly from the concrete copy \(Q_8\subset \mathbb {H}^{\times }\) in the quaternion group inside Hamilton's Quaternions, rather than assumed from a classification theorem.

Let \(u_q\) denote the basis element of the Group Algebra indexed by \(q\in Q_8\), and write a general element as \(x=\sum _{q\in Q_8}a_qu_q\). The four characters of \(Q_8/\{\pm 1\}\cong C_2\times C_2\) and the quaternionic map \(u_q\mapsto q\) combine to an algebra homomorphism \[ \Phi :\mathbb {R}[Q_8]\longrightarrow \mathbb {R}^4\times \mathbb {H}. \] For \(q\in \{1,i,j,k\}\), put \(s_q=a_q+a_{-q}\) and \(d_q=a_q-a_{-q}\). The quaternion coordinate of \(\Phi \) recovers \[ d_1+d_i i+d_j j+d_k k. \] The four real coordinates recover the Hadamard transform \[ s_1+\epsilon s_i+\delta s_j+\epsilon \delta s_k, \qquad \epsilon ,\delta \in \{\pm 1\}. \] The Hadamard matrix is invertible, so these four values recover every \(s_q\). Together with the four differences, they recover every coefficient. Thus \(\Phi \) has zero kernel. Both sides have real dimension 8, so \(\Phi \) is an isomorphism. The quaternionic realization and multiplication used here are developed in Hamilton's Quaternions and [voight2021quaternion, sec. 11.2, p. 166].

Taking centers now gives \[ Z\bigl (\mathbb {R}[Q_8]\bigr ) \cong Z\bigl (\mathbb {R}^4\times \mathbb {H}\bigr ) \cong \mathbb {R}^5. \] The comparison is: \[ \begin {array}{c|c|c} \text {object}&\text {retained data}&\text {forgotten data}\\ \hline \mathbb {R}[Q_8]&4\mathbb {R}\text { and }\mathbb {H}&\text {none here}\\ Z(\mathbb {R}[Q_8])&\text {five central factors}& \mathbb {H}\text { and block size or type}. \end {array} \] For a real-centered simple block, the center alone cannot distinguish matrix size or real from quaternionic type. A complex block still has complex center. Hence \(\operatorname {Spec}Z(\mathbb {R}[Q_8])\) is a selected commutative shadow, not a lossless localization of the noncommutative algebra.

Let \(C_2=\{1,s\}\) act on \(X=\{-1,0,1\}\) by \(s\mathbin {\cdot }x=-x\). The action groupoid \(C_2\ltimes X\) has the points of \(X\) as objects and an arrow \((h,x):x\to h\mathbin {\cdot }x\) for every \(h\in C_2\). Omitting identity arrows, its shape is

The coarse quotient remembers only the two orbits:

The retained information is already the isotropy arrow in the action groupoid. Passing to its presheaf category organizes families of such data, but does not create that arrow. This test supports the groupoid bridge and does not, by itself, admit a topos layer.

Let \(G\) be a group object in a topos \(\mathcal {E}\). A left \(G\)-object \(T\) is a torsor when \(T\to 1\) is an epimorphism and \[ (\mu ,\pi _2):G\times T\longrightarrow T\times T, \qquad (g,t)\longmapsto (g\mathbin {\cdot }t,t) \] is an isomorphism. The second condition is the internal form of freeness and transitivity. See [maclane1992sheaves, sec. VIII.2, pp. 429--430].

For an ordinary group \(G\), write \[ \mathsf {B}G=\mathsf {Set}^{BG^{\mathrm {op}}} \] for the topos of right \(G\)-sets, regarded as presheafs on the one-object category \(BG\). Its universal torsor \(U_G\) has underlying right \(G\)-set \(G\) with regular right multiplication. The constant group object \(\underline {G}\), whose right \(G\)-action is trivial, acts on \(U_G\) by left multiplication. This supplies the torsor action; the left and right actions commute.

For \(C_2\), the regular left action is a torsor in \(\mathsf {Set}\): \[ C_2\times C_2\longrightarrow C_2\times C_2, \qquad (g,h)\longmapsto (gh,h) \] is a bijection. The trivial action on a 2-point set \(D\), despite \(D\to 1\) being onto, is not a torsor. Its corresponding map sends \((g,d)\) to \((d,d)\), so it is neither injective nor surjective. Inhabitation and cardinality alone do not supply a torsor.

Let \(G\) be a group and \(\mathcal {E}\) a topos over \(\mathsf {Set}\). There is a natural equivalence \[ \operatorname {Geom}_{/\mathsf {Set}}(\mathcal {E},\mathsf {B}G) \simeq \operatorname {Tor}(\mathcal {E},G) \] between geometric morphisms over \(\mathsf {Set}\) to \(\mathsf {B}G\) and \(G\)-torsors in \(\mathcal {E}\). Under this equivalence, a geometric morphism \[ f:\mathcal {E}\longrightarrow \mathsf {B}G \] classifies the torsor \(f^*U_G\).

This is Theorem VIII.2.7 of [maclane1992sheaves]; its theorem and proof are on printed pp. 431--433.

The two directions are visible in the diagram

For a single finite action, freeness and transitivity can be checked without topoi. The added value here is one universal object classifying torsors in every topos over \(\mathsf {Set}\) and carrying them along inverse image.

For a geometric morphism \[ f:\mathcal {E}\longrightarrow \mathcal {F}, \] we write \[ f^*:\mathcal {F}\longrightarrow \mathcal {E}, \qquad f_*:\mathcal {E}\longrightarrow \mathcal {F}, \qquad f^*\dashv f_*. \] Thus the morphism and its inverse-image functor point in opposite directions. The functor \(f^*\) preserves finite limits.

Two examples fix the convention: \[ \begin {array}{c|c|c} \text {input map}&\text {geometric morphism}&\text {inverse image}\\ \hline i:\{x\}\hookrightarrow X& \mathsf {Set}\to \mathsf {Sh}(X)& i^*F=F_x=\displaystyle \varinjlim _{x\in U}F(U)\\[3pt] \varphi :H\to G& \mathsf {B}H\to \mathsf {B}G& \operatorname {Res}^G_H:\mathsf {B}G\to \mathsf {B}H. \end {array} \] In the first row, the stalk is the filtered colimit of all neighborhood sections, not the value on one chosen neighborhood.

Mac Lane and Moerdijk construct inverse-image sheaves through pulled-back étale spaces in [maclane1992sheaves, sec. II.9], especially printed pp. 99--101.

In the second row, right actions are presheaves. Precomposition with \(B\varphi ^{\mathrm {op}}\) is restriction of actions. It has both Kan-extension adjoints, so it is the inverse-image part of the displayed geometric morphism. The homomorphism and geometric morphism point from \(H\) to \(G\); restriction points from \(G\)-objects to \(H\)-objects.

Work in \(\mathsf {B}C_2\), the topos of right \(C_2\)-sets, and let \(U=C_2\) carry the regular right action. The unique equivariant map \(U\to 1\) is an epimorphism because its underlying function is surjective. In the internal language, this says that \(U\) is inhabited. Internal Group Actions are developed in [maclane1992sheaves, sec. V.2, pp. 237--240].

A global element would be an equivariant map \(1\to U\). Its value would have to satisfy \(u\mathbin {\cdot }s=u\), but the regular action has no fixed point. Hence \[ \operatorname {Hom}_{\mathsf {B}C_2}(1,U)=\varnothing . \] The two external tests are: \[ \begin {array}{c|c} \text {internal statement}&\text {external test}\\ \hline U\text { is inhabited}&U\to 1\text { is epi}\\ U\text { has a global point}&U^{C_2}\neq \varnothing . \end {array} \]

More generally, for a nontrivial group \(G\), the regular right \(G\)-object \(U_G=G\) satisfies \[ U_G\to 1\text { is epi}, \qquad \Gamma (U_G)=U_G^G=\varnothing . \] Indeed, if \(xg=x\) for every \(g\), cancellation forces every \(g\) to be \(1\). Internal existence is therefore generalized or local existence; it does not choose a global invariant element. This distinction is not a physical existence claim.

The finite tests separate calculations that merely take place in a topos from calculations that use topos-level structure: \[ \begin {array}{c|c|c|c} \text {test}&\text {lower level}&\text {topos value}&\text {verdict}\\ \hline C_3\text { action}&\text {linear action}&\text {none}&\text {fail}\\ C_2\ltimes X&\text {isotropy}&\text {none}&\text {fail}\\ C_2\text { torsor}&\text {free/transitive}&\text {classification}&\text {conditional}\\ \text {context}&\text {pullback/restriction}&\text {inverse image}&\text {conditional}\\ U_G&\text {no fixed point}&\text {internal inhabitation}&\text {strict}. \end {array} \]

Topoi enter these notes only when a result uses universal torsor classification, organizes several constructions as inverse image or base change, or relies on internal reasoning that differs from global sections. Ordinary Group Actions, Group Algebra calculations, and action groupoids stay below that layer.

This boundary also excludes several stronger claims. The spectrum of a center is not the original noncommutative Group Algebra. The action-groupoid example does not require a quotient stack. No site, descent theorem, or physical correspondence has been supplied by these finite tests. Each of those would need its own mathematical construction before extending the present layer.

The same binary tetrahedral symmetry now has several models. They answer different questions.

• The concrete model \(T\leq \mathbb {H}^{\times }\) retains quaternion coordinates, multiplication, conjugation, and norm. It is the model for explicit calculations.
• The internal semidirect-product structure retains the same group \(T\), but also marks the subgroups \(Q,C\leq T\). The facts \[ T=QC,\qquad Q\mathrel {\trianglelefteq }T,\qquad Q\cap C=\{1\} \] say that every element of \(T\) has a unique factorization \(qc\). They expose how the two parts fit inside the concrete group.
• The external model \(Q\rtimes _{\alpha }C\) replaces a quaternion by a pair \((q,c)\). It retains the factors and records their interaction in the conjugation action \(\alpha (c)(q)=cqc^{-1}\). Its multiplication is \[ (q,c)(q',c') =\bigl (q\alpha (c)(q'),cc'\bigr ). \]
• An abstract model can specify the binary tetrahedral group without quaternion coordinates. This is convenient when only its group structure is needed, but coordinates and named factors must be restored by additional maps.

The multiplication map \[ \mu :Q\rtimes _{\alpha }C\longrightarrow T,\qquad (q,c)\longmapsto qc \] is an isomorphism by the internal semidirect-product theorem. Indeed, \[ \mu \bigl ((q,c)(q',c')\bigr ) =q(cq'c^{-1})cc' =qcq'c' =\mu (q,c)\mu (q',c'). \] Thus the external multiplication is not an analogy: it is exactly the concrete multiplication written in factor coordinates. The two carriers are still different, so a theorem or construction passes between them only through an explicit isomorphism. If it refers to the named factors or their action, the transport must also record the corresponding compatibility.

The internal and external constructions are developed in [fre2023discrete, sec. 4.2.13, pp. 62--63] and [isaev2018theory, sec. 1.4.2, pp. 58--61]. The concrete identification of \(2T\) with the Hurwitz unit group and \(Q_8\rtimes \mathbb {Z}/3\mathbb {Z}\) is given in [voight2021quaternion, sec. 11.2.4, p. 168].

The semidirect-product model and the quaternion model open two different routes from the same group. One rewrites the Group Algebra in interacting factor data. The other places the group inside \(\mathrm {SU}(2)\) and leads to the McKay graph.