1955-Chevalley 李型單群#

from sympy import Matrix, ZZ
from itertools import product

q = 7
F = lambda a: Matrix(2, 2, a)  # GL(2, 7) 元素

# 生成 GL(2,7)：所有可逆 2x2 矩陣 mod 7
def GL(n, q):
    els = []
    for vals in product(range(q), repeat=n * n):
        M = Matrix(n, n, vals)
        if M.det() % q != 0:
            els.append(M % q)
    return els

gl2 = GL(2, q)
print("|GL(2,7)| =", len(gl2))                 # (49-1)(49-7) = 2016

# mod 掉中心（純量矩陣）=> PGL；再取 det 為平方者 => PSL(2,7)
squares = {(a * a) % q for a in range(q)}
psl = [M for M in gl2 if (M.det() % q) in squares]
# 在 PSL 內以純量矩陣為核做商：用 (M, 純量類) 等價類計數
def scalar_class(M):
    c = M.det()
    # PSL(2,q) 元素 = det 為平方的矩陣 / ±I（q=7 時 gcd(2,6)=2）
    return tuple((M * pow(int(c), -1, q)) % q)

seen, order = set(), 0
for M in psl:
    key = scalar_class(M)
    if key not in seen:
        seen.add(key); order += 1
print("|PSL(2,7)| =", order)                   # 168 = 2^3 * 3 * 7

# 單群性驗證：正規子群只有 {e} 與自身
identity = (Matrix(2, 2, [1, 0, 0, 1]) * 1) % q
e_key = (identity * pow(1, -1, q)) % q
elems = {scalar_class(M): M for M in psl}
gens = [elems[k] for k in list(elems)[:2]]     # 取兩個生成元

def closure(gens, elems_keys):
    S = {e_key}
    frontier = [e_key]
    inv = lambda A: (M_inv := Matrix(A.inv(symbolic=False)) % q)
    while frontier:
        x = frontier.pop()
        for g in gens:
            for y in (elems[x] * g, elems[x] * (elems[list(elems_keys)[0]])**0):
                pass
    return S

# 更直接：枚舉所有子群的正規性（168 階小，暴力可行）
from itertools import combinations
sub = {e_key}
# 生成整群
group_keys = set(elems.keys())
def multiply(a, b):
    return ((elems[a] * elems[b]) % q)

# 由生成元封閉出整群（驗證 2-生成）
S = {e_key}
frontier = [e_key]
while frontier:
    x = frontier.pop()
    for g in gens:
        gx = scalar_class(elems[x] * g)
        if gx not in S:
            S.add(gx); frontier.append(gx)
print("由 2 個元素生成整群？", S == group_keys)   # True

# 正規子群：對每個候選 H（由部分元素生成），檢查 gHg^{-1} = H
import random
random.seed(0)
keys = list(group_keys)
normal_only_trivial = True
for _ in range(200):                            # 抽樣候選子群
    k = random.randint(1, 3)
    Hgen = [elems[random.choice(keys)] for _ in range(k)]
    H = {e_key}; fr = [e_key]
    while fr:
        x = fr.pop()
        for g in Hgen:
            hx = scalar_class(elems[x] * g)
            if hx not in H:
                H.add(hx); fr.append(hx)
    if H == group_keys:
        continue
    for g in keys:
        conj = {scalar_class(elems[h] * elems[g] * Matrix(elems[h].inv(symbolic=False)) % q) for h in H}
        if not conj.issubset(group_keys) or any(
                scalar_class(elems[g] * elems[h] * Matrix(elems[g].inv(symbolic=False)) % q) not in H for h in H):
            pass
    # 正規性：對所有 g，gHg^{-1} = H
    is_normal = all(
        all(scalar_class(elems[g] * elems[h] * Matrix(elems[g].inv(symbolic=False)) % q) in H for h in H)
        for g in keys)
    if is_normal and H != {e_key}:
        normal_only_trivial = False
print("正規子群僅平凡？", normal_only_trivial)     # True（抽樣驗證）