Find all subgroups of S4 of 4,and decide which ones are normal

把集合S4的所有子组找出来，再判断它们中哪些是正规子组：S4是四元素的全置换群，它的子组有： (1) 单位元素； (2) 全置换群S4； (3) 轴对称群D4； (4) 对称群C4； (5) 群V； (6) 群D； (7) 群C； (8) 对称群A4； 其中，(1)，(2)，(3)，(8)都是正规子组。

相关问题

Kars is tired and resentful of the narrow mindset of his village since they are content with staying where they are and are not trying to become the perfect life form. Being a top-notch inventor, Kars wishes to enhance his body and become the perfect life form. Unfortunately, n of the villagers have become suspicious of his ideas. The i -th villager has a suspicion of ai on him. Individually each villager is scared of Kars, so they form into groups to be more powerful. The power of the group of villagers from l to r be defined as f(l,r) where f(l,r)=|al−al+1|+|al+1−al+2|+…+|ar−1−ar|. Here |x−y| is the absolute value of x−y . A group with only one villager has a power of 0 . Kars wants to break the villagers into exactly k contiguous subgroups so that the sum of their power is minimized. Formally, he must find k−1 positive integers 1≤r1<r2<…<rk−1<n such that f(1,r1)+f(r1+1,r2)+…+f(rk−1+1,n) is minimised. Help Kars in finding the minimum value of f(1,r1)+f(r1+1,r2)+…+f(rk−1+1,n) . Input The first line contains a single integer t (1≤t≤100) — the number of test cases. The description of test cases follows. The first line of each test case contains two integers n,k (1≤k≤n≤100) — the number of villagers and the number of groups they must be split into. The second line of each test case contains n integers a1,a2,…,an (1≤ai≤500) — the suspicion of each of the villagers. Output For each test case, output a single integer — the minimum possible value of sum of power of all the groups i. e. the minimum possible value of f(1,r1)+f(r1+1,r2)+…+f(rk−1+1,n) . Example inputCopy 3 4 2 1 3 5 2 6 3 1 9 12 4 7 2 12 8 1 9 8 2 3 3 1 8 7 7 9 2 outputCopy 4 11 2 Note In the first test case, we will group the villagers with suspicion (1,3,5,2) into (1,3,5) and (2) . So, f(1,3)+f(4,4)=(|1−3|+|3−5|)+0=4+0=4 . In the second test case, we will group the villagers with suspicion (1,9,12,4,7,2) into (1),(9,12),(4,7,2) . So, f(1,1)+f(2,3)+f(4,6)=0+3+8=11 .

这是一个关于分组问题的题目，题目给出了每个村民的怀疑程度，要求将村民分成 k 个连续的子组，使得他们的总怀疑程度最小。我们可以使用动态规划来解决这个问题。

首先，我们定义一个二维数组 dp，其中 dp[i][j] 表示将前 i 个村民分成 j 个连续子组时的最小怀疑程度。

接下来，我们可以使用状态转移方程来计算 dp[i][j]。假设当前村民为第 i 个，我们可以考虑将其与前面的村民组成一个子组，或者单独作为一个子组。如果将其与前面的村民组成一个子组，则 dp[i][j] = dp[k][j-1] + f(k+1, i)，其中 k < i。如果单独作为一个子组，则 dp[i][j] = dp[i-1][j-1] + f(i, i)，即当前村民的怀疑程度。我们取这两种情况中的最小值作为 dp[i][j] 的值。

最后，答案就是 dp[n][k]。

下面是一个具体的实现示例：

#include <iostream>
#include <vector>
#include <cmath>
using namespace std;

int solve(vector<int>& suspicions, int n, int k) {
    vector<vector<int>> dp(n + 1, vector<int>(k + 1, 0));
    vector<vector<int>> prefixSum(n + 1, vector<int>(n + 1, 0));

    for (int i = 1; i <= n; i++) {
        prefixSum[i][i] = suspicions[i - 1];
        for (int j = i + 1; j <= n; j++) {
            prefixSum[i][j] = prefixSum[i][j - 1] + suspicions[j - 1];
        }
    }

    for (int i = 1; i <= n; i++) {
        dp[i][1] = prefixSum[1][i];
    }

    for (int j = 2; j <= k; j++) {
        for (int i = j; i <= n; i++) {
            dp[i][j] = INT_MAX;
            for (int l = j - 1; l < i; l++) {
                dp[i][j] = min(dp[i][j], dp[l][j - 1] + prefixSum[l + 1][i]);
            }
        }
    }

    return dp[n][k];
}

int main() {
    int t;
    cin >> t;

    while (t--) {
        int n, k;
        cin >> n >> k;

        vector<int> suspicions(n);
        for (int i = 0; i < n; i++) {
            cin >> suspicions[i];
        }

        int result = solve(suspicions, n, k);
        cout << result << endl;
    }

    return 0;
}

希望这个解答对你有帮助！如果还有其他问题，请随时问我。

NVIDIA Vulkan更新后，开发者如何利用新特性Subgroups优化图形渲染性能？

在NVIDIA的Vulkan更新中，Subgroups提供了一种高效的跨线程通信原语，这对于在图形着色器中实现更高效的线程间同步和数据交换至关重要。开发者可以通过以下步骤来利用Subgroups优化图形渲染性能：首先，确保你的着色器代码是针对支持Subgroups的硬件编写的，比如NVIDIA的SM6.0及更高版本的GPU。接下来，你可以使用Vulkan的Shader库，它提供了访问Subgroups的接口。在着色器代码中，利用Subgroups内置函数来执行跨线程的数据交换、归约以及子组级别的同步操作。例如，你可以使用` subgroupElect`来实现屏障同步，或者使用` subgroupBroadcast`来在子组成员间广播数据。通过这种方式，你可以减少对共享内存的依赖，从而降低延迟并提升渲染效率。此外，你还可以使用NVIDIA提供的其他扩展和工具来进一步优化性能，例如使用EXT_sampler_filter_minmax扩展来提高体素化算法和稀疏数据处理的性能。为了更好地理解和应用这些技术，建议查看《NVIDIA Vulkan 更新：低级图形API的最新进展》一书，书中详细介绍了这些新特性及其应用实践。

参考资源链接：[NVIDIA Vulkan 更新：低级图形API的最新进展](https://wenku.csdn.net/doc/8173hpgd26?spm=1055.2569.3001.10343)