阶的话貌似不能比m大吧
在mod下面的剩余类最多也就是m个。在通常意义下的指数运算在这个剩余类集合上是封闭的？所以……
求指正。

那个群的运算是？  如果*运算代表模加法的话,结论是显然的 = =

不会的，阶最大为 n-1

阶n只是次数，并不是元素啊。。。

元素可加多次-----无穷啊

“模剰余类中群元素”这个说法有点别扭。你要表达的意思是不是：既约剩余系中元素的阶能比模大吗？

看到一个相等的,
Z6*  ord(5) = 6

我群语言不熟练。。。

模剰余类元素不一定素数，求阶算乘法，点阶能比模大吗？。。。

既约剩余系是元素是素数， 求阶算幂，如果模也是素数P，群阶=P-1，可点阶能比模大吗？。。。

ECC中HESSEN定理的点阶可以比P大啊，不过运算不同。。。

循环群有结果，交换群不知怎样

要是数大了。。。

还有个问题，会不会有限群有无限的点阶啊

给定一个正整数M，模M同余的整数组成的集合称为剩余类；所以剩余类有M个，由这M个剩余类组成的集合称为完全剩余系；可以证明，任一剩余类中所有元素与模数的最大公约数都是相同的，所以可以定义剩余类与模数的最大公约数，当然也就可以定义剩余类与模数互素；与M互素的剩余类组成的集合称为既约剩余系；可以证明，既约剩余系构成一个有限交换群，该群的阶正好就是欧拉函数；通常密码中所说的与模相关的群就是这个群；通常的加减乘幂，都可以在完全剩余系中定义，但问题是在完全剩余系中，任给一个系数k和剩余类P，不一定存在剩余类G，使得kG=P；但在既约剩余系中，就一定存在唯一的G使得kG=P；既约剩余系实际上可以理解为在完全剩余系中挑出一部分特殊元素，使其满足“除法”运算；实际上RSA的公私钥为什么是mod VN(欧拉函数)而不是mod N互逆的，以及ElGamal 机制中离散对数的构造，都是利用的既约剩余系。
“模剩余类中群元素”不知道你所说的群是指哪个集合以及怎么定义的运算。
ECC中的点阶和曲线参数的p的大小关系与你的问题一点关系都没有。
另外，有限群中任一元素的阶一定是有限的，这是显然的，因为若某一元素的阶无限，则由该元素可以导出一个无限子群，矛盾。

谢！

完全剩余系Zm,

既约剩余系U(m),
完全剩余系中，交换加群，所以，每个元素自加或理解成分别用1，2,3,4。。。相乘等于e,如果是Zm群，e=0

Zm中每个元素阶,只须最大乘m就能成保正模后=   e=0，虽然不一定是阶（最小正整数）

既约剩余系U(m),      e=1  群的阶  欧拉函数    =4
U(10),      ( 1-,3-,7-9-)MOD 10     ,e=1-,这四个系都有阶1,4,4,2

要是m大了,既约剩余系U(10000) 能保正=   e=-1吗？

还有多项式剩余系能保正吗？

拉格朗日定理的推论给了结果,只要是有限群：对任意的a∈G,ord(a)是|G|的因子

试试看：
G := Sym({0.. 100});
G;
R:=Order(G);
R;
Factorization(R);
a:=Random(G);
a;

a1:=a^100;
a1;
a2:=a^300;
a3:=a^10000;
a3;

orda1:=Order(a1);
orda1;
orda2:=Order(a2);
orda2;
orda3:=Order(a3);
orda3;
Gcd(R,orda1);
Gcd(R,orda2);
Gcd(R,orda3);

Symmetric group G acting on a set of cardinality 101
9425947759838359420851623124482936749562312794702543768327889353416977599316221\
4765030878615918083469116234900035495995833697063026032640000000000000000000000\
00
[ <2, 97>, <3, 48>, <5, 24>, <7, 16>, <11, 9>, <13, 7>, <17, 5>, <19, 5>, <23,
4>, <29, 3>, <31, 3>, <37, 2>, <41, 2>, <43, 2>, <47, 2>, <53, 1>, <59, 1>, <61,
1>, <67, 1>, <71, 1>, <73, 1>, <79, 1>, <83, 1>, <89, 1>, <97, 1>, <101, 1> ]
(0, 89, 46, 93, 28, 5, 3, 58, 69, 83, 34, 68, 51, 37, 39, 95, 2, 42, 64, 43, 50,
    81, 61, 53, 33, 70, 87, 24, 1, 36, 17, 16, 38, 57, 74, 100, 52, 35, 56, 63,
    65, 23, 44, 71, 96, 85, 26, 6)(4, 14, 59, 66)(7, 82, 55, 10, 15, 62, 29, 86,
    13, 30, 45, 78, 32, 8, 79, 67, 99, 48, 25, 11, 94, 60, 80, 12, 27, 54, 9,
    19, 92, 77, 49, 73, 40, 90)(18, 84, 41, 97, 88, 21, 75, 20, 47, 31, 72, 98,
    76, 22)
(0, 28, 69, 51, 2, 50, 33, 1, 38, 52, 65, 96)(3, 34, 39, 64, 61, 87, 17, 74, 56,
    44, 26, 46)(5, 83, 37, 42, 81, 70, 36, 57, 35, 23, 85, 89)(6, 93, 58, 68,
    95, 43, 53, 24, 16, 100, 63, 71)(7, 40, 49, 92, 9, 27, 80, 94, 25, 99, 79,
    32, 45, 13, 29, 15, 55)(8, 78, 30, 86, 62, 10, 82, 90, 73, 77, 19, 54, 12,
    60, 11, 48, 67)(18, 41, 88, 75, 47, 72, 76)(20, 31, 98, 22, 84, 97, 21)
(0, 2, 38)(1, 96, 51)(3, 61, 56)(5, 81, 35)(6, 95, 16)(7, 15, 13, 32, 99, 94,
    27, 92, 40, 55, 29, 45, 79, 25, 80, 9, 49)(8, 48, 60, 54, 77, 90, 10, 86,
    78, 67, 11, 12, 19, 73, 82, 62, 30)(17, 26, 39)(18, 88, 47, 76, 41, 75,
    72)(20, 98, 84, 21, 31, 22, 97)(23, 83, 70)(24, 71, 68)(28, 50, 52)(33, 65,
    69)(34, 87, 44)(36, 85, 37)(42, 57, 89)(43, 100, 93)(46, 64, 74)(53, 63, 58)
1428
476
357
1428
476
357

G := Sym({ "a", "b", "c", "d" ,"e", "f", "g", "h" ,"i", "j", "k", "l","m" });

G;
R:=Order(G);
R;
Factorization(R);
a:=Random(G);
a;

a1:=a^100;
a1;
a2:=a^300;
a3:=a^10000;
a3;
a4:=a^1000450000;
a4;

orda1:=Order(a1);
orda1;
orda2:=Order(a2);
orda2;
orda3:=Order(a3);
orda3;
orda4:=Order(a4);
orda4;

Gcd(R,orda1);
Gcd(R,orda2);
Gcd(R,orda3);
Gcd(R,orda4);

Symmetric group G acting on a set of cardinality 13
Order = 2^10 * 3^5 * 5^2 * 7 * 11 * 13
6227020800
[ <2, 10>, <3, 5>, <5, 2>, <7, 1>, <11, 1>, <13, 1> ]
(f, i)(b, g, m, a, k, c, e, l, h)
(b, g, m, a, k, c, e, l, h)
(b, g, m, a, k, c, e, l, h)
(b, g, m, a, k, c, e, l, h)
9
3
9
9
9
3
9
9

西罗P/西罗子P   /ABELSUB/所有的子群,子孙生的真多,S7就运行不了了，S5才行

验拉格朗日定理：
left:= a1 * G;
left;
right:=G * a1;
right;
quot:=a1*a3 * G;

G := Sym({ 0..6 });

G;
R:=Order(G);
R;
Factorization(R);
FactoredOrder(G);
a:=Random(G);
a;

a1:=a^10;
a1;
a2:=a^300;
a3:=a^10000;
a3;
a4:=a^1000450000;
a4;

orda1:=Order(a1);
orda1;
orda2:=Order(a2);
orda2;
orda3:=Order(a3);
orda3;
orda4:=Order(a4);
orda4;

Gcd(R,orda1);
Gcd(R,orda2);
Gcd(R,orda3);
Gcd(R,orda4);

SylowSubgroup(G, 2);
Sylow(G, 2);
SylowSubgroup(G, 3);
Sylow(G, 3);
SylowSubgroup(G, 173);
Sylow(G, 173);

AbelianSubgroups(G);

Symmetric group G acting on a set of cardinality 7
Order = 5040 = 2^4 * 3^2 * 5 * 7
5040
[ <2, 4>, <3, 2>, <5, 1>, <7, 1> ]
[ <2, 4>, <3, 2>, <5, 1>, <7, 1> ]
(0, 2, 6, 1)
(0, 6)(1, 2)
Id(G)
Id(G)
2
1
1
1
2
1
1
1
Permutation group acting on a set of cardinality 7
Order = 16 = 2^4
    (0, 1)
    (2, 3)
    (2, 4)(3, 5)
Permutation group acting on a set of cardinality 7
Order = 16 = 2^4
    (0, 1)
    (2, 3)
    (2, 4)(3, 5)
Permutation group acting on a set of cardinality 7
Order = 9 = 3^2
    (0, 1, 2)
    (3, 4, 5)
Permutation group acting on a set of cardinality 7
Order = 9 = 3^2
    (0, 1, 2)
    (3, 4, 5)
Permutation group acting on a set of cardinality 7
Order = 1
Permutation group acting on a set of cardinality 7
Order = 1
Conjugacy classes of subgroups
------------------------------

[ 1]    Order 1            Length 1
        Permutation group acting on a set of cardinality 7
        Order = 1
            Id($)
[ 2]    Order 2            Length 21
        Permutation group acting on a set of cardinality 7
        Order = 2
            (5, 6)
[ 3]    Order 2            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 2
            (0, 4)(1, 3)
[ 4]    Order 2            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 2
            (0, 1)(3, 6)(4, 5)
[ 5]    Order 3            Length 35
        Permutation group acting on a set of cardinality 7
        Order = 3
            (1, 4, 3)
[ 6]    Order 3            Length 140
        Permutation group acting on a set of cardinality 7
        Order = 3
            (0, 6, 4)(1, 2, 3)
[ 7]    Order 5            Length 126
        Permutation group acting on a set of cardinality 7
        Order = 5
            (0, 1, 2, 4, 3)
[ 8]    Order 7            Length 120
        Permutation group acting on a set of cardinality 7
        Order = 7
            (0, 3, 6, 2, 4, 5, 1)
[ 9]    Order 4            Length 35
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (0, 4)(1, 3)
            (0, 1)(3, 4)
[10]    Order 4            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (0, 5, 1, 6)
            (0, 1)(5, 6)
[11]    Order 4            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (2, 6)(3, 4)
            (0, 1)(2, 6)
[12]    Order 4            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (5, 6)
            (0, 1)(5, 6)
[13]    Order 4            Length 315
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (0, 2, 6, 4)(1, 3)
            (0, 6)(2, 4)
[14]    Order 4            Length 315
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (0, 3)(1, 4)(5, 6)
            (5, 6)
[15]    Order 4            Length 315
        Permutation group acting on a set of cardinality 7
        Order = 4 = 2^2
            (0, 1)(3, 6)(4, 5)
            (3, 5)(4, 6)
[16]    Order 6            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 6 = 2 * 3
            (3, 4, 5)
            (0, 1)(2, 6)
[17]    Order 6            Length 210
        Permutation group acting on a set of cardinality 7
        Order = 6 = 2 * 3
            (0, 1, 4)
            (5, 6)
[18]    Order 6            Length 420
        Permutation group acting on a set of cardinality 7
        Order = 6 = 2 * 3
            (0, 3, 5)(1, 6, 4)
            (0, 1)(3, 6)(4, 5)
[19]    Order 9            Length 70
        Permutation group acting on a set of cardinality 7
        Order = 9 = 3^2
            (2, 3, 4)
            (0, 1, 6)
[20]    Order 10           Length 126
        Permutation group acting on a set of cardinality 7
        Order = 10 = 2 * 5
            (0, 2, 1, 6, 3)
            (4, 5)
[21]    Order 8            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 8 = 2^3
            (0, 3)(1, 4)(5, 6)
            (0, 1)(3, 4)
            (0, 1)(3, 4)(5, 6)
[22]    Order 8            Length 105
        Permutation group acting on a set of cardinality 7
        Order = 8 = 2^3
            (3, 4)(5, 6)
            (0, 1)(3, 4)
            (0, 1)(3, 4)(5, 6)
[23]    Order 8            Length 315
        Permutation group acting on a set of cardinality 7
        Order = 8 = 2^3
            (0, 4, 1, 3)
            (0, 1)(3, 4)
            (0, 1)(3, 4)(5, 6)
[24]    Order 12           Length 35
        Permutation group acting on a set of cardinality 7
        Order = 12 = 2^2 * 3
            (3, 4, 5)
            (0, 2)(1, 6)
            (0, 6)(1, 2)
[25]    Order 12           Length 105
        Permutation group acting on a set of cardinality 7
        Order = 12 = 2^2 * 3
            (2, 3, 6)
            (0, 1)(4, 5)
            (4, 5)
[26]    Order 12           Length 105
        Permutation group acting on a set of cardinality 7
        Order = 12 = 2^2 * 3
            (0, 5, 6, 4)
            (1, 3, 2)
            (0, 6)(4, 5)

西罗P/西罗子P   /ABELSUB也看看

Symmetric group G acting on a set of cardinality 5
Order = 120 = 2^3 * 3 * 5
120
[ <2, 3>, <3, 1>, <5, 1> ]
[ <2, 3>, <3, 1>, <5, 1> ]
(0, 2)(1, 4)
Id(G)
Id(G)
Id(G)
1
1
1
1
1
1
1
1
Permutation group acting on a set of cardinality 5
Order = 8 = 2^3
    (0, 1)
    (0, 2)(1, 3)
Permutation group acting on a set of cardinality 5
Order = 8 = 2^3
    (0, 1)
    (0, 2)(1, 3)
Permutation group acting on a set of cardinality 5
Order = 3
    (0, 2, 3)
Permutation group acting on a set of cardinality 5
Order = 3
    (0, 2, 3)
Permutation group acting on a set of cardinality 5
Order = 1
Permutation group acting on a set of cardinality 5
Order = 1
Conjugacy classes of subgroups
------------------------------

[ 1]    Order 1            Length 1
        Permutation group acting on a set of cardinality 5
        Order = 1
            Id($)
[ 2]    Order 2            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 2
            (2, 3)
[ 3]    Order 2            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 2
            (0, 1)(2, 3)
[ 4]    Order 3            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 3
            (1, 2, 3)
[ 5]    Order 5            Length 6
        Permutation group acting on a set of cardinality 5
        Order = 5
            (0, 3, 4, 1, 2)
[ 6]    Order 4            Length 5
        Permutation group acting on a set of cardinality 5
        Order = 4 = 2^2
            (0, 1)(2, 3)
            (0, 3)(1, 2)
[ 7]    Order 4            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 4 = 2^2
            (0, 3, 4, 2)
            (0, 4)(2, 3)
[ 8]    Order 4            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 4 = 2^2
            (2, 3)
            (0, 4)(2, 3)
[ 9]    Order 6            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 6 = 2 * 3
            (0, 1, 4)
            (2, 3)
Conjugacy classes of subgroups
------------------------------

[ 1]    Order 1            Length 1
        Permutation group acting on a set of cardinality 5
        Order = 1
            Id($)
[ 2]    Order 2            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 2
            (3, 4)
[ 3]    Order 2            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 2
            (0, 1)(3, 4)
[ 4]    Order 3            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 3
            (1, 4, 3)
[ 5]    Order 5            Length 6
        Permutation group acting on a set of cardinality 5
        Order = 5
            (0, 3, 2, 1, 4)
[ 6]    Order 4            Length 5
        Permutation group acting on a set of cardinality 5
        Order = 4 = 2^2
            (0, 1)(3, 4)
            (0, 3)(1, 4)
[ 7]    Order 4            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 4 = 2^2
            (0, 3, 2, 4)
            (0, 2)(3, 4)
[ 8]    Order 4            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 4 = 2^2
            (3, 4)
            (0, 2)(3, 4)
[ 9]    Order 6            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 6 = 2 * 3
            (1, 4)(2, 3)
            (0, 4, 1)
[10]    Order 6            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 6 = 2 * 3
            (0, 1, 2)
            (3, 4)
[11]    Order 6            Length 10
        Permutation group acting on a set of cardinality 5
        Order = 6 = 2 * 3
            (3, 4)
            (2, 4, 3)
[12]    Order 10           Length 6
        Permutation group acting on a set of cardinality 5
        Order = 10 = 2 * 5
            (1, 2)(3, 4)
            (0, 3, 2, 1, 4)
[13]    Order 8            Length 15
        Permutation group acting on a set of cardinality 5
        Order = 8 = 2^3
            (3, 4)
            (0, 2)(3, 4)
            (0, 3)(2, 4)
[14]    Order 12           Length 5
        Permutation group acting on a set of cardinality 5
        Order = 12 = 2^2 * 3
            (1, 4, 3)
            (0, 1)(3, 4)
            (0, 3)(1, 4)
[15]    Order 12           Length 10
        Permutation group acting on a set of cardinality 5
        Order = 12 = 2^2 * 3
            (1, 2)
            (0, 1, 2)
            (3, 4)
[16]    Order 20           Length 6
        Permutation group acting on a set of cardinality 5
        Order = 20 = 2^2 * 5
            (0, 3, 4, 2)
            (0, 4)(2, 3)
            (0, 3, 1, 2, 4)
[17]    Order 24           Length 5
        Permutation group acting on a set of cardinality 5
        Order = 24 = 2^3 * 3
            (3, 4)
            (2, 4, 3)
            (0, 2)(3, 4)
            (0, 3)(2, 4)
[18]    Order 60           Length 1
        Permutation group acting on a set of cardinality 5
        Order = 60 = 2^2 * 3 * 5
            (0, 1)(2, 3)
            (1, 4, 2)
[19]    Order 120          Length 1
        Permutation group acting on a set of cardinality 5
        Order = 120 = 2^3 * 3 * 5
            (0, 1)
            (0, 4)(1, 2, 3)

可以去研究下有限交换群的基本定理（有限阿贝尔群基本定理），这个对理解有限交换群至关重要，对于密码分析也很有帮助。

看了N次了，感觉就是素数拆分到最小再组合一下排列，可那只是结构，和循环群混着用的时后就老咪糊，不只是不是还有有限交换群的2级基本定理，比基本定理高点的