【初めてのJuliaプログラミング】とある有限群を具体的に求めてみた。

1109

こちらは「日曜数学 Advent Calendar 2020 」の記事です。
素敵な力作がたくさん集まってますので、ぜひ読んでみてください。

Introduction

以前、Mathlogさんに３日前に群を知ったねこという記事を書き、円周群を取り上げました。

集合$C$を以下とする。
$$ C=\{(x,y)|x,y \in \mathbb{R},\ x^2+y^2=1\} $$

$C$の任意の元$(x_1,y_1),(x_2,y_2)$に対して、ある実数$\theta_1,\theta_2$が存在して
$$ (x_1,y_1)=(\cos\theta_1,\sin\theta_1)\\ (x_2,y_2)=(\cos\theta_2,\sin\theta_2)\\ $$
と表せる。
演算$\cdot$を以下のように定義する。
$$ (x_1,y_1)\cdot(x_2,y_2)=(\cos(\theta_1+\theta_2),\sin(\theta_1+\theta_2)) $$
このとき、$C$は演算$\cdot$に関して可換群をなす。

こちらを実数でなく$\mathbb{Z}/p\mathbb{Z}$にしてみると、以下が成り立ちます。

集合$C_p$を以下とする。
$$ C_p=\{(\overline{x},\overline{y})|\overline{x},\overline{y} \in \mathbb{Z}/p\mathbb{Z},\ \overline{x}^2+\overline{y}^2=\overline{1}\} $$

演算$\cdot$を以下のように定義する。
$$ (\overline{x_1},\overline{y_1})\cdot(\overline{x_2},\overline{y_2})=(\overline{x_1}\cdot\overline{x_2}-\overline{y_1}\cdot\overline{y_2},\ \overline{x_1}\cdot\overline{y_2}+\overline{y_1}\cdot\overline{x_2}) $$
このとき、$C_p$は演算$\cdot$に関して可換群をなす。

$\mathbb{R}$の場合をお手本にしながら、三角関数の加法定理をヒントにして演算を定義しています。
こちらの証明は、のちのち３日前に群を知ったねこの続きとして、ねこさんに証明してもらいたいので、今回は省略します。

この記事では、具体的に$C_p$の元を求め、どんな群になっているかを見ていきます。

まずは手計算で

$p=2$

$$ C_2=\{(\overline{x},\overline{y})|\overline{x},\overline{y} \in \mathbb{Z}/2\mathbb{Z},\ \overline{x}^2+\overline{y}^2=\overline{1}\} $$

$(\overline{x},\overline{y})$を$C_2$の元とする。

$\overline{x}=\overline{0}$とすると、$\overline{y}^2=\overline{1}$より、$\overline{y}=\overline{1}$

$\overline{x}=\overline{1}$とすると、$\overline{y}^2=\overline{0}$より、$\overline{y}=\overline{0}$

よって
$$ C_2 = \{(\overline{0},\overline{1}),(\overline{1},\overline{0})\} $$
ここで
$$ (\overline{0},\overline{1})\cdot(\overline{0},\overline{1})=(\overline{0}\cdot\overline{0}-\overline{1}\cdot\overline{1},\ \overline{0}\cdot\overline{1}+\overline{1}\cdot\overline{0})=(\overline{1},\overline{0}) $$
となるので
$$ C_2 = \{(\overline{0},\overline{1}),(\overline{0},\overline{1})^2\} $$

よって、$C_2$は位数$2$の巡回群となる。

$p=3$

$$ C_3=\{(\overline{x},\overline{y})|\overline{x},\overline{y} \in \mathbb{Z}/3\mathbb{Z},\ \overline{x}^2+\overline{y}^2=\overline{1}\} $$

$(\overline{x},\overline{y})$を$C_3$の元とする。

$\overline{x}=\overline{0}$とすると、$\overline{y}^2=\overline{1}$より、$\overline{y}=\overline{\pm 1}$

$\overline{x}=\overline{1}$とすると、$\overline{y}^2=\overline{0}$より、$\overline{y}=\overline{0}$

$\overline{x}=\overline{-1}$とすると、$\overline{y}^2=\overline{0}$より、$\overline{y}=\overline{0}$

よって
$$ C_3 = \{(\overline{0},\overline{1}),(\overline{0},\overline{-1}),(\overline{1},\overline{0}),(\overline{-1},\overline{0})\} $$
ここで
$$ \begin{align} (\overline{0},\overline{1})\cdot(\overline{0},\overline{1})&=(\overline{0}\cdot\overline{0}-\overline{1}\cdot\overline{1},\ \overline{0}\cdot\overline{1}+\overline{1}\cdot\overline{0})\\ &=(\overline{-1},\overline{0}) \end{align} $$

$$ \begin{align} (\overline{0},\overline{1})\cdot(\overline{0},\overline{1})\cdot(\overline{0},\overline{1})&=(\overline{-1},\overline{0})\cdot(\overline{0},\overline{1})\\ &=(\overline{-1}\cdot\overline{0}-\overline{0}\cdot\overline{1},\ \overline{-1}\cdot\overline{1}+\overline{0}\cdot\overline{0})\\ &=(\overline{0},\overline{-1}) \end{align} $$

$$ \begin{align} (\overline{0},\overline{1})\cdot(\overline{0},\overline{1})\cdot(\overline{0},\overline{1})\cdot(\overline{0},\overline{1})&=(\overline{0},\overline{-1})\cdot(\overline{0},\overline{1})\\ &=(\overline{0}\cdot\overline{0}-\overline{(-1)}\cdot\overline{1},\ \overline{0}\cdot\overline{1}+\overline{(-1)}\cdot\overline{0})\\ &=(\overline{1},\overline{0}) \end{align} $$
となるので
$$ C_3 = \{(\overline{0},\overline{1}),(\overline{0},\overline{1})^2,(\overline{0},\overline{1})^3,(\overline{0},\overline{1})^4\} $$

よって、$C_3$は位数$4$の巡回群となる。

お願い！Juliaさん！

手計算が疲れてきたので、文明の利器に頼ります。丸美屋の麻婆豆腐を作ることよりも環境構築が簡単なことで有名なプログラミング言語 Julia で書きました。
AbstractAlgebra.jl というスーパー便利そうなパッケージがありますが、初心者すぎるため、今回は Primes.jl だけ使って愚直に書いてしまいました。また今度コーディングするときは、AbstractAlgebra.jlを使ってみようと思います。

計算結果

$2< p<10^2$での計算結果です。以下の順に記載しています。

$p$
$C_p$の位数
$C_p$の元（上線はついていません）

$C_p$の元については、一つ目に記載したものが生成元となっており、二つ目がその$2$乗、三つ目がその$3$乗、四つ目がその$4$乗・・・といった具合になっています。
また、位数が大きいものについては、スクロールすることで全ての元を見ることができます。

※私なりに確認はしていますが、初心者なので書いたプログラムにミスがあるかもしれません。以下の結果を真面目に参照する場合は、ぜひご自身でもご確認頂ければと思います。

$p\equiv 1 \pmod 4$

$p=5$
位数:$4$

$$ [(0, 1), (4, 0), (0, 4), (1, 0)] $$

$p=13$
位数:$12$

$$ [(2, 6), (7, 11), (0, 12), (6, 11), (11, 6), (12, 0), (11, 7), (6, 2), (0, 1), (7, 2), (2, 7), (1, 0)] $$

$p=17$
位数:$16$

$$ [(4, 6), (14, 14), (6, 4), (0, 1), (11, 4), (3, 14), (13, 6), (16, 0), (13, 11), (3, 3), (11, 13), (0, 16), (6, 13), (14, 3), (4, 11), (1, 0)] $$

$p=29$
位数:$28$

$$ [(5, 11), (20, 23), (21, 16), (16, 21), (23, 20), (11, 5), (0, 1), (18, 5), (6, 20), (13, 21), (8, 16), (9, 23), (24, 11), (28, 0), (24, 18), (9, 6), (8, 13), (13, 8), (6, 9), (18, 24), (0, 28), (11, 24), (23, 9), (16, 8), (21, 13), (20, 6), (5, 18), (1, 0)] $$

$p=37$
位数:$36$

$$ [(2, 16), (7, 27), (26, 18), (23, 8), (29, 14), (19, 11), (10, 30), (21, 35), (0, 36), (16, 35), (27, 30), (18, 11), (8, 14), (14, 8), (11, 18), (30, 27), (35, 16), (36, 0), (35, 21), (30, 10), (11, 19), (14, 29), (8, 23), (18, 26), (27, 7), (16, 2), (0, 1), (21, 2), (10, 7), (19, 26), (29, 23), (23, 29), (26, 19), (7, 10), (2, 21), (1, 0)] $$

$p=41$
位数:$40$

$$ [(13, 18), (9, 17), (16, 14), (38, 19), (29, 29), (19, 38), (14, 16), (17, 9), (18, 13), (0, 1), (23, 13), (24, 9), (27, 16), (22, 38), (12, 29), (3, 19), (25, 14), (32, 17), (28, 18), (40, 0), (28, 23), (32, 24), (25, 27), (3, 22), (12, 12), (22, 3), (27, 25), (24, 32), (23, 28), (0, 40), (18, 28), (17, 32), (14, 25), (19, 3), (29, 12), (38, 22), (16, 27), (9, 24), (13, 23), (1, 0)] $$

$p=53$
位数:$52$

$$ [(5, 20), (49, 41), (8, 19), (31, 43), (37, 40), (21, 39), (14, 32), (13, 16), (10, 22), (34, 45), (12, 4), (33, 48), (0, 52), (20, 48), (41, 4), (19, 45), (43, 22), (40, 16), (39, 32), (32, 39), (16, 40), (22, 43), (45, 19), (4, 41), (48, 20), (52, 0), (48, 33), (4, 12), (45, 34), (22, 10), (16, 13), (32, 14), (39, 21), (40, 37), (43, 31), (19, 8), (41, 49), (20, 5), (0, 1), (33, 5), (12, 49), (34, 8), (10, 31), (13, 37), (14, 21), (21, 14), (37, 13), (31, 10), (8, 34), (49, 12), (5, 33), (1, 0)] $$

$p=61$
位数:$60$

$$ [(2, 27), (7, 47), (26, 39), (36, 48), (57, 31), (9, 15), (40, 29), (29, 40), (15, 9), (31, 57), (48, 36), (39, 26), (47, 7), (27, 2), (0, 1), (34, 2), (14, 7), (22, 26), (13, 36), (30, 57), (46, 9), (32, 40), (21, 29), (52, 15), (4, 31), (25, 48), (35, 39), (54, 47), (59, 27), (60, 0), (59, 34), (54, 14), (35, 22), (25, 13), (4, 30), (52, 46), (21, 32), (32, 21), (46, 52), (30, 4), (13, 25), (22, 35), (14, 54), (34, 59), (0, 60), (27, 59), (47, 54), (39, 35), (48, 25), (31, 4), (15, 52), (29, 21), (40, 32), (9, 46), (57, 30), (36, 13), (26, 22), (7, 14), (2, 34), (1, 0)] $$

$p=73$
位数:$72$

$$ [(6, 29), (71, 56), (43, 59), (7, 68), (41, 27), (47, 37), (12, 52), (24, 3), (57, 57), (3, 24), (52, 12), (37, 47), (27, 41), (68, 7), (59, 43), (56, 71), (29, 6), (0, 1), (44, 6), (17, 71), (14, 43), (5, 7), (46, 41), (36, 47), (21, 12), (70, 24), (16, 57), (49, 3), (61, 52), (26, 37), (32, 27), (66, 68), (30, 59), (2, 56), (67, 29), (72, 0), (67, 44), (2, 17), (30, 14), (66, 5), (32, 46), (26, 36), (61, 21), (49, 70), (16, 16), (70, 49), (21, 61), (36, 26), (46, 32), (5, 66), (14, 30), (17, 2), (44, 67), (0, 72), (29, 67), (56, 2), (59, 30), (68, 66), (27, 32), (37, 26), (52, 61), (3, 49), (57, 16), (24, 70), (12, 21), (47, 36), (41, 46), (7, 5), (43, 14), (71, 17), (6, 44), (1, 0)] $$

$p=89$
位数:$88$

$$ [(13, 30), (70, 68), (27, 47), (9, 86), (29, 53), (33, 46), (28, 75), (72, 35), (64, 34), (79, 48), (32, 57), (41, 10), (55, 25), (54, 17), (14, 61), (43, 56), (36, 60), (3, 80), (42, 62), (21, 19), (59, 76), (0, 88), (30, 76), (68, 19), (47, 62), (86, 80), (53, 60), (46, 56), (75, 61), (35, 17), (34, 25), (48, 10), (57, 57), (10, 48), (25, 34), (17, 35), (61, 75), (56, 46), (60, 53), (80, 86), (62, 47), (19, 68), (76, 30), (88, 0), (76, 59), (19, 21), (62, 42), (80, 3), (60, 36), (56, 43), (61, 14), (17, 54), (25, 55), (10, 41), (57, 32), (48, 79), (34, 64), (35, 72), (75, 28), (46, 33), (53, 29), (86, 9), (47, 27), (68, 70), (30, 13), (0, 1), (59, 13), (21, 70), (42, 27), (3, 9), (36, 29), (43, 33), (14, 28), (54, 72), (55, 64), (41, 79), (32, 32), (79, 41), (64, 55), (72, 54), (28, 14), (33, 43), (29, 36), (9, 3), (27, 42), (70, 21), (13, 59), (1, 0)] $$

$p=97$
位数:$96$

$$ [(14, 22), (3, 34), (70, 57), (17, 10), (18, 29), (2, 26), (38, 20), (92, 49), (16, 91), (65, 74), (58, 41), (7, 7), (41, 58), (74, 65), (91, 16), (49, 92), (20, 38), (26, 2), (29, 18), (10, 17), (57, 70), (34, 3), (22, 14), (0, 1), (75, 14), (63, 3), (40, 70), (87, 17), (68, 18), (71, 2), (77, 38), (48, 92), (6, 16), (23, 65), (56, 58), (90, 7), (39, 41), (32, 74), (81, 91), (5, 49), (59, 20), (95, 26), (79, 29), (80, 10), (27, 57), (94, 34), (83, 22), (96, 0), (83, 75), (94, 63), (27, 40), (80, 87), (79, 68), (95, 71), (59, 77), (5, 48), (81, 6), (32, 23), (39, 56), (90, 90), (56, 39), (23, 32), (6, 81), (48, 5), (77, 59), (71, 95), (68, 79), (87, 80), (40, 27), (63, 94), (75, 83), (0, 96), (22, 83), (34, 94), (57, 27), (10, 80), (29, 79), (26, 95), (20, 59), (49, 5), (91, 81), (74, 32), (41, 39), (7, 90), (58, 56), (65, 23), (16, 6), (92, 48), (38, 77), (2, 71), (18, 68), (17, 87), (70, 40), (3, 63), (14, 75), (1, 0)] $$

$p\equiv 3 \pmod 4$

$p=3$
位数:$4$

$$ [(0, 1), (2, 0), (0, 2), (1, 0)] $$

$p=7$
位数:$8$

$$ [(2, 2), (0, 1), (5, 2), (6, 0), (5, 5), (0, 6), (2, 5), (1, 0)] $$

$ p=11$
位数:$12$

$$ [(3, 5), (6, 8), (0, 10), (5, 8), (8, 5), (10, 0), (8, 6), (5, 3), (0, 1), (6, 3), (3, 6), (1, 0)] $$

$p=19$
位数:$20$

$$ [(3, 7), (17, 4), (4, 17), (7, 3), (0, 1), (12, 3), (15, 17), (2, 4), (16, 7), (18, 0), (16, 12), (2, 15), (15, 2), (12, 16), (0, 18), (7, 16), (4, 2), (17, 15), (3, 12), (1, 0)] $$

$ p=23$
位数:$24$

$$ [(4, 10), (8, 11), (14, 9), (12, 15), (13, 19), (0, 22), (10, 19), (11, 15), (9, 9), (15, 11), (19, 10), (22, 0), (19, 13), (15, 12), (9, 14), (11, 8), (10, 4), (0, 1), (13, 4), (12, 8), (14, 14), (8, 12), (4, 13), (1, 0)] $$

$p=31$
位数:$32$

$$ [(2, 11), (7, 13), (26, 10), (4, 27), (21, 5), (18, 24), (20, 29), (0, 30), (11, 29), (13, 24), (10, 5), (27, 27), (5, 10), (24, 13), (29, 11), (30, 0), (29, 20), (24, 18), (5, 21), (27, 4), (10, 26), (13, 7), (11, 2), (0, 1), (20, 2), (18, 7), (21, 26), (4, 4), (26, 21), (7, 18), (2, 20), (1, 0)] $$

$p=43$
位数:$44$

$$ [(3, 11), (17, 23), (13, 41), (18, 8), (9, 7), (36, 34), (35, 25), (2, 30), (20, 26), (32, 40), (0, 42), (11, 40), (23, 26), (41, 30), (8, 25), (7, 34), (34, 7), (25, 8), (30, 41), (26, 23), (40, 11), (42, 0), (40, 32), (26, 20), (30, 2), (25, 35), (34, 36), (7, 9), (8, 18), (41, 13), (23, 17), (11, 3), (0, 1), (32, 3), (20, 17), (2, 13), (35, 18), (36, 9), (9, 36), (18, 35), (13, 2), (17, 20), (3, 32), (1, 0)] $$

$p=47$
位数:$48$

$$ [(4, 19), (31, 11), (9, 22), (41, 24), (37, 29), (20, 20), (29, 37), (24, 41), (22, 9), (11, 31), (19, 4), (0, 1), (28, 4), (36, 31), (25, 9), (23, 41), (18, 37), (27, 20), (10, 29), (6, 24), (38, 22), (16, 11), (43, 19), (46, 0), (43, 28), (16, 36), (38, 25), (6, 23), (10, 18), (27, 27), (18, 10), (23, 6), (25, 38), (36, 16), (28, 43), (0, 46), (19, 43), (11, 16), (22, 38), (24, 6), (29, 10), (20, 27), (37, 18), (41, 23), (9, 25), (31, 36), (4, 28), (1, 0)] $$

$p=59$
位数:$60$

$$ [(11, 23), (5, 34), (40, 17), (49, 45), (35, 29), (13, 3), (15, 37), (22, 44), (56, 46), (30, 24), (14, 10), (42, 19), (25, 54), (36, 48), (0, 58), (23, 48), (34, 54), (17, 19), (45, 10), (29, 24), (3, 46), (37, 44), (44, 37), (46, 3), (24, 29), (10, 45), (19, 17), (54, 34), (48, 23), (58, 0), (48, 36), (54, 25), (19, 42), (10, 14), (24, 30), (46, 56), (44, 22), (37, 15), (3, 13), (29, 35), (45, 49), (17, 40), (34, 5), (23, 11), (0, 1), (36, 11), (25, 5), (42, 40), (14, 49), (30, 35), (56, 13), (22, 15), (15, 22), (13, 56), (35, 30), (49, 14), (40, 42), (5, 25), (11, 36), (1, 0)] $$

$p=67$
位数:$68$

$$ [(3, 27), (17, 28), (32, 7), (41, 14), (13, 10), (37, 46), (8, 65), (11, 9), (58, 56), (2, 59), (21, 30), (57, 54), (53, 26), (60, 35), (39, 50), (40, 64), (0, 66), (27, 64), (28, 50), (7, 35), (14, 26), (10, 54), (46, 30), (65, 59), (9, 56), (56, 9), (59, 65), (30, 46), (54, 10), (26, 14), (35, 7), (50, 28), (64, 27), (66, 0), (64, 40), (50, 39), (35, 60), (26, 53), (54, 57), (30, 21), (59, 2), (56, 58), (9, 11), (65, 8), (46, 37), (10, 13), (14, 41), (7, 32), (28, 17), (27, 3), (0, 1), (40, 3), (39, 17), (60, 32), (53, 41), (57, 13), (21, 37), (2, 8), (58, 11), (11, 58), (8, 2), (37, 21), (13, 57), (41, 53), (32, 60), (17, 39), (3, 40), (1, 0)] $$

$ p=71$
位数:$72$

$$ [(13, 20), (53, 23), (16, 10), (8, 24), (50, 46), (14, 36), (30, 38), (56, 29), (6, 6), (29, 56), (38, 30), (36, 14), (46, 50), (24, 8), (10, 16), (23, 53), (20, 13), (0, 1), (51, 13), (48, 53), (61, 16), (47, 8), (25, 50), (35, 14), (33, 30), (42, 56), (65, 6), (15, 29), (41, 38), (57, 36), (21, 46), (63, 24), (55, 10), (18, 23), (58, 20), (70, 0), (58, 51), (18, 48), (55, 61), (63, 47), (21, 25), (57, 35), (41, 33), (15, 42), (65, 65), (42, 15), (33, 41), (35, 57), (25, 21), (47, 63), (61, 55), (48, 18), (51, 58), (0, 70), (20, 58), (23, 18), (10, 55), (24, 63), (46, 21), (36, 57), (38, 41), (29, 15), (6, 65), (56, 42), (30, 33), (14, 35), (50, 25), (8, 47), (16, 61), (53, 48), (13, 51), (1, 0)] $$

$ p=79$
位数:$80$

$$ [(2, 32), (7, 49), (26, 6), (18, 54), (46, 52), (8, 75), (65, 11), (15, 48), (74, 23), (44, 44), (23, 74), (48, 15), (11, 65), (75, 8), (52, 46), (54, 18), (6, 26), (49, 7), (32, 2), (0, 1), (47, 2), (30, 7), (73, 26), (25, 18), (27, 46), (4, 8), (68, 65), (31, 15), (56, 74), (35, 44), (5, 23), (64, 48), (14, 11), (71, 75), (33, 52), (61, 54), (53, 6), (72, 49), (77, 32), (78, 0), (77, 47), (72, 30), (53, 73), (61, 25), (33, 27), (71, 4), (14, 68), (64, 31), (5, 56), (35, 35), (56, 5), (31, 64), (68, 14), (4, 71), (27, 33), (25, 61), (73, 53), (30, 72), (47, 77), (0, 78), (32, 77), (49, 72), (6, 53), (54, 61), (52, 33), (75, 71), (11, 14), (48, 64), (23, 5), (44, 35), (74, 56), (15, 31), (65, 68), (8, 4), (46, 27), (18, 25), (26, 73), (7, 30), (2, 47), (1, 0)] $$

$p=83$
位数:$84$

$$ [(3, 18), (17, 25), (16, 49), (79, 20), (43, 71), (13, 74), (35, 41), (31, 6), (68, 78), (45, 47), (36, 38), (5, 15), (77, 52), (42, 48), (9, 70), (12, 40), (63, 4), (34, 67), (58, 66), (65, 80), (0, 82), (18, 80), (25, 66), (49, 67), (20, 4), (71, 40), (74, 70), (41, 48), (6, 52), (78, 15), (47, 38), (38, 47), (15, 78), (52, 6), (48, 41), (70, 74), (40, 71), (4, 20), (67, 49), (66, 25), (80, 18), (82, 0), (80, 65), (66, 58), (67, 34), (4, 63), (40, 12), (70, 9), (48, 42), (52, 77), (15, 5), (38, 36), (47, 45), (78, 68), (6, 31), (41, 35), (74, 13), (71, 43), (20, 79), (49, 16), (25, 17), (18, 3), (0, 1), (65, 3), (58, 17), (34, 16), (63, 79), (12, 43), (9, 13), (42, 35), (77, 31), (5, 68), (36, 45), (45, 36), (68, 5), (31, 77), (35, 42), (13, 9), (43, 12), (79, 63), (16, 34), (17, 58), (3, 65), (1, 0)] $$