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Mathematical Derivations, continued

Parts: A, B, C, D, E.

13. Sporadic Group Orders and Triangular Decomposition

13.1. The $2$-adic Valuations $v_{2}(|Co_{1}|)=v_{2}(|Fi_{24}^{\prime}|)=21$

The Conway group $Co_1$ (Conway group Co1) has order $$|Co_1| ;=; 2^{21} \cdot 3^{9} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 23.$$ This factors as $$|Co_1| ;=; 4,157,776,806,543,360,000.$$ The $2$-adic valuation of this order is exactly $21$: $$v_2(|Co_1|) ;=; 21.$$ Direct verification: $$|Co_1| / 2^{21} ;=; \frac{4,157,776,806,543,360,000}{2,097,152} ;=; 1,982,582,476,875,$$ which is odd, and in fact divisible by $3, 5, 7, 11, 13, 23$. Therefore $v_2(|Co_1|) = 21$ is verified.

The Fischer group $Fi_{24}'$ (the derived subgroup of $Fi_{24}$) has order $$|Fi_{24}'| ;=; 2^{21} \cdot 3^{16} \cdot 5^{2} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29.$$ This also has $v_2 = 21$. The verification follows the same structure.

Among the $26$ sporadic simple groups (sporadic group), only $Co_1$ and $Fi_{24}'$ have $v_2(|G|) = 21$. The other sporadic groups have $v_2(|G|) \in \{3, 6, 7, 8, 9, 10, 14, 15, 17, 18, 41, 46\}$. Notably:

Group $v_2(\lvert G\rvert)$
$M_{11}$ $6$
$M_{12}$ $6$
$J_1$ $3$
McL $7$
$HS$ $9$
$Fi_{22}$ $17$
$Fi_{23}$ $18$
$\mathbf{Fi_{24}^{\prime}}$ $\mathbf{21}$
$Co_1$ $\mathbf{21}$
$Co_2$ $18$
$Co_3$ $10$
$HN$ $14$
$Th$ $15$
$B$ $41$
$M$ $46$

Connection to $21 = 3 \cdot 7$. The order of $Co_1$ has the structure $|Co_1| = 2^{21} \cdot 3^9 \cdot 7^2 \cdot (\text{other primes})$. Therefore $21 = 3 \cdot 7$ divides $|Co_1|$, with $3^9$ and $7^2$ as separate factors. The $2$-adic valuation $21$ encodes $21$ as the largest power of $2$ dividing the group order. The largest $2$-power, $2^{21}$, is itself a nontrivial object.

The connection to the central node is two-fold:

  1. $21$ divides $|Co_1|$ (the order has $21$ as a factor).
  2. $21 = v_2(|Co_1|)$ (the $2$-adic valuation equals $21$).

These are independent facts about the same number.

13.2. The Sylow Structure of $Co_{1}$

The order $|Co_1| = 2^{21} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23$ has the following Sylow consequences (Sylow theorems):

  • A Sylow $2$-subgroup has order $2^{21}$.
  • A Sylow $7$-subgroup has order $7^2 = 49$.
  • The number of Sylow $7$-subgroups is congruent to $1$ modulo $7$ and divides $|Co_1|/49$.

The exponent $v_7(|Co_1|) = 2$ (not cubed or higher) means $Co_1$ contains elements of order $49$ but not $343$.

13.3. Uniqueness of $21=T_{3}+T_{5}$ as a Sum of Two Triangular Numbers

By Gauss's Eureka theorem, every positive integer is a sum of at most three triangular numbers. For $N = 21$, the unique nontrivial pair $(T_a, T_b)$ with $1 \le a < b$ and $T_a + T_b = 21$ is: $$21 ;=; T_3 + T_5 ;=; 6 + 15.$$ Proof by direct enumeration. The equation $T_a + T_b = 21$ is equivalent to $a(a+1) + b(b+1) = 42$ with $1 \leq a < b$. Checking $a \in {1, 2, 3, 4}$:

$a$ $b(b+1) = 42 - a(a+1)$ Integer $b$?
$1$ $40$ No ($6 \times 7 = 42$)
$2$ $36$ No ($5 \times 6 = 30$, $6 \times 7 = 42$)
$3$ $30$ $b = 5$ yes
$4$ $22$ No ($4 \times 5 = 20$, $5 \times 6 = 30$)

Therefore $(a, b) = (3, 5)$ is the unique solution, giving $21 = T_3 + T_5 = 6 + 15$.

Remark. The pair $(3, 5)$ satisfies $3 + 5 = 8 = F_6$ and $3 \cdot 5 = 15 = T_5$. The general identity $T_n = T_a + T_b$ requires $n(n+1) = a(a+1) + b(b+1)$.

14. Euler Characteristic of $\mathbb{CP}^{20}$ and Additional Intersections

14.1. Euler Characteristic of $\mathbb{CP}^{20}$ equals $21$

The complex projective space $\mathbb{CP}^{n}$ is the quotient of $\mathbb{C}^{n+1} \setminus {0}$ by $\mathbb{C}^{\times}$ (complex projective space). It has complex dimension $n$, real dimension $2n$. The cohomology ring is $$H^{*}(\mathbb{CP}^{n}; \mathbb{Z}) = \mathbb{Z}[x]/(x^{n+1}), \quad \deg x = 2.$$ The Betti numbers are $$b_{2k}(\mathbb{CP}^{n}) = 1 \quad \text{for } 0 \le k \le n, \qquad b_{j}(\mathbb{CP}^{n}) = 0 \text{ otherwise}.$$ The Euler characteristic is $$\chi(\mathbb{CP}^{n}) = \sum_{k=0}^{n} b_{2k} = n + 1.$$ At $n = 20$: $$\chi(\mathbb{CP}^{20}) = 21.$$ The space $\mathbb{CP}^{20}$ has $21$ nontrivial even-degree cohomology groups, each $\mathbb{Z}$, with $21$ Betti numbers summing to $21$.

14.2. Two Groups of Order $21$

The number of groups of order $n$ is tabulated in OEIS A000001. For $n \le 30$ the values are $$1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, \mathbf{2}, 2, 1, 15, 2, 2, 5, 4, 1, 4.$$ At $n = 21$, the count is exactly $2$:

  1. Cyclic group $C_{21} = \mathbb{Z}/21\mathbb{Z}$ (abelian).
  2. Frobenius group $F_{21} = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$ (non-abelian).

The non-trivial action of $\mathbb{Z}/3$ on $\mathbb{Z}/7$ uses the cubic roots of unity modulo $7$, namely $k \in {2, 4}$ (since $k^3 \equiv 1 \pmod 7$). Choosing $k = 2$ gives $a b a^{-1} = b^2$, whose orbit of $b$ is ${b, b^2, b^4}$ (order $3$, consistent with $|a| = 3$). The choice $k = 4$ gives the isomorphic group with $a b a^{-1} = b^4$.

Conjugacy classes. $F_{21}$ has exactly $5$ conjugacy classes, of sizes $1, 3, 3, 7, 7$:

  • ${1}$ (identity),
  • ${b, b^2, b^4}$ and ${b^3, b^5, b^6}$ (orbits of $\mathbb{Z}/7 \setminus {1}$ under $a$),
  • ${(b^i, a) : i \in \mathbb{Z}/7}$ and ${(b^i, a^2) : i \in \mathbb{Z}/7}$ (size $7$ each).

Irreducible representations. The $5$ irreps have dimensions $1, 1, 1, 3, 3$, satisfying $\sum d_i^2 = 1 + 1 + 1 + 9 + 9 = 21 = |F_{21}|$:

  • Two linear reps from the abelian quotient $F_{21} \to \mathbb{Z}/3$ (trivial and sign of $\mathbb{Z}/3$).
  • One linear rep from the trivial character of the normal subgroup $\mathbb{Z}/7$.
  • Two degree-$3$ irreps from the induced representations $\mathrm{Ind}_{\mathbb{Z}/7}^{F_{21}}(\text{trivial})$ and $\mathrm{Ind}_{\mathbb{Z}/7}^{F_{21}}(\text{sign})$.

Sources

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