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TheMatrix 写了： 2022年 12月 25日 15:06 这里用了点symmetric group的知识。Symmetric group就是permutation （function）on N objects。写出来就是

YWY 写了： 2022年 12月 24日 21:07 There are only 3 conjugacy classes in A_5, one containing identity (only), one containing (1 2)(3 4), while the other containing (1 2 3). Here we write a permutation as a product of disjoint cycles.

The conjugacy class containing (1 2) and the class containing (1 2 3)(4 5) all consist of odd permutations. So yes, there are altogether 5 conjugacy classes in S_5.

For some silly reason, I missed the conjugacy class of (1 2 3 4), which is odd, and the class of (1 2 3 4 5), which is even. So, in S_5, there are 7 conjugacy classes, of which 4 classes are even.

TheMatrix 写了： 2022年 12月 24日 17:07

TheMatrix 写了： 2022年 12月 25日 15:06 这里用了点symmetric group的知识。Symmetric group就是permutation （function）on N objects。写出来就是

(0,1,2,3,4) - 单位元
(3,2,0,1,4) - 看成函数的话就是 (0->3, 1->2, 2->0, 3->1, 4->4)

这样的。共有N!个。可以相乘 - 就是函数的合成， 构成了群。

第一个函数 SymmN，就是写出N个元素的全部permutation。这个小问题放在LeetCode上，肯定不是easy的，至少是medium的。

第二个函数 times，就是permutation的乘法，也就是函数的合成。这在LeetCode上应该也是medium的。

第三个函数 inverse，就是一个permutation的inverse。这也得是medium的。

第四个函数conjugate，就是乘法 hgh^-1。用乘法函数就简单了。

第五个函数 subgroup，是一个permutation的集合生成的子群，也就是以集合中元素各种乘法张成的子群。这个在LeetCode里应该是hard。

最后一个函数 conj_closure，是一个子集在全集中的全部conjugate。

[(0,), (1,), (2,), (3,), (4,)]

（ヅ） 写了： 2022年 12月 26日 12:14 老哥你别再用这色调了可以吗

根本看不见

verdelite 写了： 2022年 12月 26日 12:21 原来是因为你的显示是黑底白字的。我的显示是白底黑字的，很清楚。你看看是不是在BBS的个人设置里面可以改。

（ヅ） 写了： 2022年 12月 26日 12:19 第一个SymmN()，我怎么吐出来这个结果?

代码： 全选

[(0,), (1,), (2,), (3,), (4,)]

N = 4

def SymmN():
    points = list(range(N))
    result = [tuple()]
    for length in range(N):
        result = [(*x, p) for x in result for p in points if p not in x]
    return result

TheMatrix 写了： 2022年 12月 26日 12:31 不应该啊。你是不是漏一行代码？

代码： 全选

N = 4

def SymmN():
    points = list(range(N))
    result = [tuple()]
    for length in range(N):
        result = [(*x, p) for x in result for p in points if p not in x]
    return result

（ヅ） 写了： 2022年 12月 26日 12:14 老哥你别再用这色调了可以吗

根本看不见

1. the conjugacy class containing the identity (only), which is even
2. the conjugacy class containing (1 2), which is odd
3. the conjugacy class containing (1 2)(3 4), which is even
4. the conjugacy class containing (1 2)(3 4)(5 6), which is odd
5. the conjugacy class containing (1 2 3), which is even
6. the conjugacy class containing (1 2 3)(4 5), which is odd
7. the conjugacy class containing (1 2 3)(4 5 6), which is even
8. the conjugacy class containing (1 2 3 4), which is odd
9. the conjugacy class containing (1 2 3 4)(5 6), which is even
10. the conjugacy class containing (1 2 3 4 5), which is even
11. the conjugacy class containing (1 2 3 4 5 6), which is odd

YWY 写了： 2022年 12月 26日 13:35 关于S_n的共轭类，我个人觉得用disjoint cycle的形式更简洁（也更简单），比如我们看S_6, the group of permutations on {1, 2, 3, 4, 5, 6}:

The conjugacy classes of S_6 are as follows:

1. the conjugacy class containing the identity (only), which is even
2. the conjugacy class containing (1 2), which is odd
3. the conjugacy class containing (1 2)(3 4), which is even
4. the conjugacy class containing (1 2)(3 4)(5 6), which is odd
5. the conjugacy class containing (1 2 3), which is even
6. the conjugacy class containing (1 2 3)(4 5), which is odd
7. the conjugacy class containing (1 2 3)(4 5 6), which is even
8. the conjugacy class containing (1 2 3 4), which is odd
9. the conjugacy class containing (1 2 3 4)(5 6), which is even
10. the conjugacy class containing (1 2 3 4 5), which is even
11. the conjugacy class containing (1 2 3 4 5 6), which is odd

Now, if you are given a permutation in S_6, say (4 2 5)(3 6), how do you know which conjugacy class it belongs to? The answer is that (4 2 5)(3 6) belongs to the conjugacy class of (1 2 3)(4 5).

Some may ask: how to list all permutations in a conjugacy class, say the conjugacy class of (1 2 3)(4 5)? The answer is that it consists of all permutations in S_6 that can be written as a product of two disjoint cycles of lengths 3 and 2 respectively, such as (1 3 2)(5 6), (2 5 4)(1 3), etc..

但愿没漏掉什么共轭类。。。总之我就是提个思路。。。大家应该已经看到了规律。

TheMatrix 写了： 2022年 12月 26日 13:53 嗯。而且disjoint cycle type不分顺序，比如
(1 2 3 4)(5 6)
(1 2)(3 4 5 6)
算同一个type？

YWY 写了： 2022年 12月 26日 14:00 对，这两个是共轭的，可以通过 1->3, 2->4, 3->5, 4->6, 5->1, 6->2来实现（就是用这个置换做共轭）。