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# Glossary cards — Chapter ᚌ(G): The hall of mirrors

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T14, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch14.md)

*Lie theory, geometric algebra (Clifford algebras), group theory, and Galois theory: brackets, bivectors, the Monster, and the unsolvable quintic. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Gort — `ᚌ` {#g-og-gort}

*Ogham letter*

Ogham letter name meaning ‘a field’; the medieval glossators forced it into the ivy.

Gort means ‘field’, and is cognate with the English word garden. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give milsiu féraib, “sweetest grass”; ined erc, “suitable place for cows”; and sásad ile, “sating of multitudes”—all three about pasture, not about a tree. The Auraicept nonetheless glosses edind, ivy: “greener than pastures is gort.” McManus names gort, with ruis and úath, as one of the letter names the medieval glossators reinterpreted as a tree epithet more or less by force. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### the Jacobi identity — `[x, [y, z]] + [y, [z, x]] + [z, [x, y]]` {#g-jacobi}

*Integer or rational*

In any Lie algebra the three nested brackets sum to 0; this is the Jacobi identity, the consistency law for the bracket.

A Lie algebra has a bilinear alternating bracket satisfying the Jacobi identity. In the equivalent derivation form, [x, [y, z]] = [[x, y], z] + [y, [x, z]]; moving every term to one side gives the cyclic zero above. The identity for Poisson brackets is in Jacobi’s Nova methodus, which Clebsch published in 1862, eleven years after Jacobi died, so the year is the edition’s and not his. Hermann Weyl introduced the name Lie algebra in 1934. Its value is the zero vector of the algebra, handed home as the integer 0.

Cites: [14.1](ch14-refs.md#r-14.1), [14.18](ch14-refs.md#r-14.18)

See also: [a Lie algebra](ch14-cards.md#g-liealg)

### the dimension of E₈ — `dim E₈` {#g-e8}

*Integer or rational*

248, the dimension of E₈, the largest of the five exceptional simple Lie algebras.

E₈ has dimension 248 and rank 8, making it the largest of the five exceptional simple Lie algebras. Its root system has 240 roots; the associated E₈ lattice gives the optimal sphere packing in eight dimensions, proved by Maryna Viazovska. An integer, 248.

Cites: [14.16](ch14-refs.md#r-14.16), [14.2](ch14-refs.md#r-14.2)

See also: [units of the integral octonions](ch23-cards.md#g-cayleyint)

### antisymmetry of the Lie bracket — `[x, x]` {#g-liealt}

*Integer or rational*

A Lie bracket of anything with itself is 0; together with bilinearity and Jacobi, that is the definition of a Lie algebra.

The definition uses bilinearity, the alternating law [x, x] = 0, and the Jacobi identity. Expanding [x + y, x + y] = 0 forces [x, y] = −[y, x]. For matrices, the commutator [x, y] = xy − yx is the standard example. The value is the zero vector, handed home as the integer 0.

Cites: [14.1](ch14-refs.md#r-14.1)

See also: [the Jacobi identity](ch14-cards.md#g-jacobi), [a Lie algebra](ch14-cards.md#g-liealg)

### the dimension of su(2) — `dim 𝔰𝔲(2)` {#g-su2}

*Integer or rational*

3: the Lie algebra of spin and of rotations, spanned by the three Pauli matrices times i.

The real Lie algebra su(2) consists of traceless anti-Hermitian 2 × 2 matrices. It is three-dimensional, with a real basis given by i times the three Pauli matrices; SU(2) double-covers SO(3). An integer, 3.

Cites: [14.3](ch14-refs.md#r-14.3)

See also: [a Lie algebra](ch14-cards.md#g-liealg), [the Pauli matrices](ch22-cards.md#g-pauli)

### the dimension of G₂ — `dim G₂` {#g-g2}

*Integer or rational*

14, the smallest exceptional Lie algebra: the symmetries of the octonions, which is why Chapters ᚌ(G) and ᚖ(W) share it.

G₂ is the smallest exceptional simple Lie algebra: it has dimension 14 and rank 2, and it is the derivation algebra of the octonions. That is the precise bridge to Chapter ᚖ(W). An integer, 14.

Cites: [14.2](ch14-refs.md#r-14.2)

See also: [a Lie algebra](ch14-cards.md#g-liealg), [the octonions](ch23-cards.md#g-octonions)

### the dimension of F₄ — `dim F₄` {#g-f4}

*Integer or rational*

52, the exceptional Lie algebra of derivations of the exceptional Jordan algebra.

F₄ has dimension 52 and rank 4. It is the derivation algebra of the exceptional Jordan algebra formed from 3 × 3 Hermitian octonionic matrices. An integer, 52.

Cites: [14.2](ch14-refs.md#r-14.2)

See also: [a Hermitian operator](ch15-cards.md#g-hermitian), [a Lie algebra](ch14-cards.md#g-liealg)

### the dimension of E₆ — `dim E₆` {#g-e6}

*Integer or rational*

78, the dimension of E₆, the rank-six exceptional simple Lie algebra.

E₆ has dimension 78 and rank 6, with a 27-dimensional fundamental representation. That representation is used in E₆ grand-unified models dating to 1976. An integer, 78.

Cites: [14.17](ch14-refs.md#r-14.17), [14.2](ch14-refs.md#r-14.2)

See also: [a Lie algebra](ch14-cards.md#g-liealg)

### the dimension of E₇ — `dim E₇` {#g-e7}

*Integer or rational*

133, the dimension of E₇, the rank-seven exceptional simple Lie algebra.

E₇ has dimension 133 and rank 7; its smallest nontrivial fundamental representation has dimension 56. An integer, 133.

Cites: [14.2](ch14-refs.md#r-14.2)

See also: [a Lie algebra](ch14-cards.md#g-liealg)

### a Lie algebra — `𝔤` {#g-liealg}

*Non-real object*

A vector space with a bracket that is bilinear and alternating and obeys the Jacobi identity: what a continuous group of symmetries looks like near the identity, with matrix commutators as the standard example.

A Lie algebra is a vector space carrying a bracket [x, y] that is bilinear and alternating, so [x, x] = 0, and that satisfies the Jacobi identity; for matrices the bracket is the commutator xy − yx. It is the infinitesimal shadow of a continuous group of symmetries—the group’s moves near the identity, with composition traded for a bracket—which is why this chapter’s first brace is written with brackets and not with the group. Over an algebraically closed field of characteristic 0 the finite-dimensional simple ones fall into the four classical series of types A–D and the five exceptional types G₂, F₄, E₆, E₇, E₈. A structure rather than a number: the chapter’s first two zeros are its zero vector.

Cites: [14.1](ch14-refs.md#r-14.1), [14.20](ch14-refs.md#r-14.20)

See also: [the dimension of E₈](ch14-cards.md#g-e8), [the dimension of G₂](ch14-cards.md#g-g2), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the Jacobi identity](ch14-cards.md#g-jacobi), [antisymmetry of the Lie bracket](ch14-cards.md#g-liealt)

### a vector wedged with itself — `e₁ ∧ e₁` {#g-wedge}

*Integer or rational*

In geometric algebra the outer product of a vector with itself is 0, because a vector spans no area with itself.

Grassmann introduced the exterior product in his 1844 Ausdehnungslehre; Clifford joined exterior and metric structure in his ~1876 algebra; the Encyclopedia of Mathematics dates the work to 1876 and other accounts to the 1878 paper, so the earliest is shown. The rule v ∧ v = 0 says that one vector spans no area with itself. Expanding (u + v) ∧ (u + v) = 0 gives u ∧ v = −v ∧ u. The value is the zero bivector, handed home as 0.

Cites: [14.4](ch14-refs.md#r-14.4)

### orthogonal vectors anticommute — `e₁e₂ + e₂e₁` {#g-cliffanti}

*Integer or rational*

In Clifford’s geometric algebra two perpendicular vectors anticommute, so this sum is 0: the rule that makes the geometric product work.

In a Clifford algebra, vectors satisfy uv + vu = 2B(u, v), where B(u, v) is zero when u and v are perpendicular, so perpendicular vectors anticommute. Thus e₁e₂ + e₂e₁ = 0. The value is the zero of the algebra, handed home as the integer 0.

Cites: [14.4](ch14-refs.md#r-14.4)

### the dimension of the geometric algebra of space — `dim Cl(3)` {#g-cl3dim}

*Integer or rational*

8 = 2³: one scalar, three vectors, three bivectors, and one trivector, every product of the three basis vectors of space.

A Clifford algebra on an n-dimensional space has a basis indexed by subsets of n basis vectors and therefore dimension 2^{n}. For n = 3 the grade dimensions are 1, 3, 3, 1, totaling 8; the even subalgebra is isomorphic to the quaternions. An integer, 8.

Cites: [14.5](ch14-refs.md#r-14.5)

See also: [the quaternions](ch22-cards.md#g-quaternions)

### the sum of the fifth roots of unity — `1 + ζ₅ + ζ₅² + ζ₅³ + ζ₅⁴` {#g-rootsum}

*Integer or rational*

The five points of a pentagon on the unit circle add to 0; the cyclic group cancels itself.

The n-th roots of unity are the vertices of a regular n-gon, and their sum is zero for n ≥ 2. In particular the five vertices shown here add to 0. Gauss’s construction of the regular 17-gon, recorded in his diary in March 1796, rested on sums of roots of unity like this one. The value is the complex number 0, handed home as an integer.

Cites: [14.19](ch14-refs.md#r-14.19), [14.6](ch14-refs.md#r-14.6)

### the order of the symmetric group on five letters — `|S₅|` {#g-s5}

*Integer or rational*

120, the first symmetric group that is not solvable, which is why the general quintic has no formula in radicals.

S₅ has 5! = 120 elements. Its normal subgroup A₅ is nonabelian simple, so S₅ is not solvable; that obstruction is why the general quintic is not solvable by radicals. An integer, 120.

Cites: [14.15](ch14-refs.md#r-14.15), [14.8](ch14-refs.md#r-14.8)

### the Monster’s smallest representation — `196883` {#g-moonshine}

*Integer or rational*

One less than the 196884 in the j-invariant; the coincidence that became monstrous moonshine.

196883 is the smallest nontrivial complex character degree of the Monster. McKay observed that 196883 + 1 = 196884, the coefficient of q in the j-function; Conway and Norton developed this into monstrous moonshine, proved by Borcherds in 1992. An integer, 196883.

Cites: [14.10](ch14-refs.md#r-14.10), [14.11](ch14-refs.md#r-14.11)

See also: [the j-invariant](ch14-cards.md#g-jinv)

### a nontrivial character sums to zero — `Σ_{g∈G} χ(g), χ ≠ 1` {#g-charsum}

*Integer or rational*

Add up any nontrivial irreducible character of a finite group over all its elements and you get 0: orthogonality of the characters Frobenius introduced in 1896.

For irreducible characters of a finite group, orthogonality with the trivial character gives Σ_{g∈G} χ(g) = 0 whenever χ is nontrivial. Frobenius introduced group characters in 1896. The value is the complex number 0, handed home as an integer.

Cites: [14.7](ch14-refs.md#r-14.7)

### the order of the alternating group on five letters — `|A₅|` {#g-a5}

*Integer or rational*

60: the smallest nonabelian simple group, the rotations of an icosahedron, and the reason the general quintic has no formula in radicals.

A₅ has 5!/2 = 60 elements. It is the rotation group of the icosahedron and the smallest nonabelian simple group; its simplicity makes S₅ nonsolvable and supplies the group-theoretic obstruction for the general quintic. An integer, 60.

Cites: [14.8](ch14-refs.md#r-14.8)

See also: [the order of the symmetric group on five letters](ch14-cards.md#g-s5)

### the number of sporadic groups — `26` {#g-sporadic26}

*Integer or rational*

Twenty-six finite simple groups belong to no infinite family; the classification theorem says there are no more.

The classification of finite simple groups has four parts: cyclic groups of prime order, alternating groups, groups of Lie type, and exactly 26 sporadic groups. The 2004 quasithin work closed the last recognized gap in the proof. An integer, 26.

Cites: [14.9](ch14-refs.md#r-14.9)

### the order of the Monster group — `|𝕄|` {#g-monster}

*Integer or rational*

808017424794512875886459904961710757005754368000000000, about 8 × 10⁵³ elements, the largest sporadic group.

The Monster has order 2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31·41·47·59·71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, about 8 × 10⁵³. It is the largest sporadic simple group, and its smallest nontrivial complex character degree is 196883. An integer.

Cites: [14.10](ch14-refs.md#r-14.10)

### the Tits group — `²F₄(2)′` {#g-titsgroup}

*Non-real object*

A finite simple group of order 17,971,200 that sits inside the Ree group ²F₄(2), classified by convention among the Lie-type families rather than counted as a twenty-seventh sporadic group.

The Tits group, named for Jacques Tits, is an index-2 subgroup of the Ree group ²F₄(2) and is simple in its own right, with order 17,971,200. It belongs to the infinite family ²F₄(2²ⁿ⁺¹)′ of derived subgroups of the Ree groups, though it is not strictly a group of Lie type itself. The classification of finite simple groups usually files it with the infinite Lie-type families rather than counting it as a sporadic group, which is why the tally of sporadic groups stays at twenty-six instead of becoming twenty-seven. Some sources call it a borderline or honorary sporadic group all the same.

Cites: [14.23](ch14-refs.md#r-14.23)

### the j-invariant — `j(τ)` {#g-jinv}

*Function or operator*

The modular j-function, with q-expansion 1/q + 744 + 196884q + …, is the basic modular function on the upper half-plane invariant under the full modular group.

The j-invariant is invariant under SL(2, ℤ) acting on the upper half-plane and generates the field of level-one modular functions; every elliptic curve over the complex numbers corresponds to a value of j, up to isomorphism. Its q-expansion, with q = e^(2πiτ), begins 1/q + 744 + 196884q + 21493760q² + …; the coefficient 196884 is one more than 196883, the dimension of the smallest nontrivial irreducible representation of the Monster group, the coincidence McKay noticed that grew into monstrous moonshine. j is also why Ramanujan’s constant e^(π√163) sits so close to an integer: at τ = (1+√−163)/2 the j-function takes the integer value −640320³, and because every term after 1/q + 744 is tiny there, e^(π√163) lies within about 10⁻¹² of 640320³ + 744.

Cites: [14.25](ch14-refs.md#r-14.25)

See also: [Ramanujan’s constant](ch04-cards.md#g-ram)

### the monstrous moonshine conjecture {#g-moonshineconj}

*Function or operator*

The Conway-Norton monstrous moonshine conjectures of 1979, proved by Borcherds in 1992, connect the coefficients of the j-function to the representations of the Monster group.

After McKay noticed that 196884, the first nontrivial coefficient of the j-invariant, is one more than 196883, the dimension of the smallest nontrivial representation of the Monster group, Conway and Norton turned the coincidence into a full conjectural correspondence in 1979: every coefficient of the modular functions attached to each element of the Monster matches a combination of the group’s representations. Richard Borcherds proved the conjectures in 1992 for the moonshine module, a vertex operator algebra with Monster symmetry, using the no-ghost theorem from string theory, work for which, in part, he received the Fields Medal in 1998.

Cites: [14.26](ch14-refs.md#r-14.26)

See also: [the j-invariant](ch14-cards.md#g-jinv)

### the discriminant of a repeated root — `disc((x − 1)²)` {#g-disc}

*Integer or rational*

b² − 4ac for x² − 2x + 1 is 4 − 4 = 0; a zero discriminant means a root repeats.

The discriminant is, up to the leading-coefficient factor, the product of the squared differences of a polynomial’s roots. For ax² + bx + c it is b² − 4ac, and it vanishes exactly when a root repeats. For (x − 1)² it is 4 − 4 = 0, an integer.

Cites: [14.12](ch14-refs.md#r-14.12)

### the degree of a field extension — `[ℚ(√2) : ℚ]` {#g-degsqrt2}

*Integer or rational*

2: adjoining √2 to the rationals doubles the dimension.

The degree [L : K] is the dimension of L as a vector space over K. Since ℚ(√2) has basis {1, √2} over ℚ, its degree is 2; extension degrees multiply in towers. An integer.

Cites: [14.13](ch14-refs.md#r-14.13)

See also: [square root of two](ch04-cards.md#g-sqrt2)

### the order of a Galois group — `|Gal(ℚ(ζ₅)/ℚ)|` {#g-galzeta5}

*Integer or rational*

4: the symmetries of the fifth roots of unity form (ℤ/5ℤ)^×, a central example in Galois theory.

For a cyclotomic field, Gal(ℚ(ζ_{n})/ℚ) is isomorphic to (ℤ/nℤ)^{×}, acting by ζ_{n} ↦ ζ_{n}^{k}. For n = 5 this group has order φ(5) = 4. An integer.

Cites: [14.14](ch14-refs.md#r-14.14)

### Abel’s bracket — `[the general quintic is solvable in radicals]` {#g-abelbracket}

*Integer or rational*

0: Abel proved in 1824 that no formula using only arithmetic and root extraction solves every fifth-degree equation, and Galois explained why: S₅ is not solvable.

Ruffini’s 1799 proof had one gap in it and Abel proved in 1824 that no formula by radicals solves every quintic. Galois theory explains the obstruction: the general quintic has group S₅, which is not solvable because it contains the nonabelian simple group A₅. The Iverson bracket of the false proposition is 0, an integer.

Cites: [14.15](ch14-refs.md#r-14.15), [14.21](ch14-refs.md#r-14.21)

See also: [Iverson bracket](ch08-cards.md#g-iverson), [the order of the symmetric group on five letters](ch14-cards.md#g-s5)