# 数学代写|抽象代数作业代写abstract algebra代考|MATH355

## 数学代写|抽象代数作业代写abstract algebra代考|The Rotation Group of a Cube and a Soccer Ball

It cannot be overemphasized that Theorem $7.4$ and Lagrange’s Theorem (Theorem 7.1) are counting theorems. ${ }^3$ They enable us to determine the numbers of elements in various sets. To see how Theorem $7.4$ works, we will determine the order of the rotation group of a cube and a soccer ball. That is, we wish to find the number of essentially different ways in which we can take a cube or a soccer ball in a certain location in space, physically rotate it, and then have it still occupy its original location.

• EXAMPLE 10 Let $G$ be the rotation group of a cube. Label the six faces of the cube 1 through 6 . Since any rotation of the cube must carry each face of the cube to exactly one other face of the cube and different rotations induce different permutations of the faces, $G$ can be viewed as a group of permutations on the set ${1,2,3,4,5,6}$. Clearly, there is some rotation about a central horizontal or vertical axis that carries face number 1 to any other face, so that $\left|\operatorname{orb}_G(1)\right|=6$. Next, we consider $\operatorname{stab}_G$ (1). Here, we are asking for all rotations of a cube that leave face number 1 where it is. Surely, there are only four such motions – rotations of $0^{\circ}, 90^{\circ}, 180^{\circ}$, and $270^{\circ}$ – about the line perpendicular to the face and passing through its center (see Figure 7.2). Thus, by Theorem $7.4,|G|=\left|\operatorname{orb}_G(1)\right|\left|\operatorname{stab}_G(1)\right|=6 \cdot 4=24$.

Now that we know how many rotations a cube has, it is simple to determine the actual structure of the rotation group of a cube. Recall that $S_4$ is the symmetric group of degree 4 .

## 数学代写|抽象代数作业代写abstract algebra代考|An Application of Cosets to the Rubik’s Cube

Recall from Chapter 5 that in 2010 it was proved via a computer computation, which took 35 CPU-years to complete, that every Rubik’s cube could be solved in at most 20 moves. To carry out this effort, the research team of Morley Davidson John Dethridge, Herbert Kociemba, and Tomas Rokicki applied a program of Rokicki, which built on early work of Kociemba, that checked the elements of the cosets of a subgroup $H$ of order $(8 ! \cdot 8 ! \cdot 4 !) / 2=$ $19,508,428,800$ to see if each cube in a position corresponding to the elements in a coset could be solved within 20 moves. In the rare cases where Rokicki’s program did not work, an alternate method was employed. Using symmetry considerations, they were able to reduce the approximately 2 billion cosets of $H$ to about 56 million cosets for testing. Cosets played a role in this effort because Rokicki’s program could handle the $19.5+$ billion elements in the same coset in about 20 seconds.

In this chapter, we show how to piece together groups to make larger groups. In Chapter 9, we will show that we can often start with one large group and decompose it into a product of smaller groups in much the same way as a composite positive integer can be broken down into a product of primes. These methods will later be used to give us a simple way to construct all finite Abelian groups.

# 抽象代数代考

## 数学代写|抽象代数作业代写abstract algebra代考|The Rotation Group of a Cube and a Soccer Ball

• 例 10 让G是一个立方体的旋转群。标记立方体 1 到 6 的六个面。由于立方体的任何旋转都必须将立方体的每个面带到立方体的另一个面，并且不同的旋转会导致面的不同排列，G可以看作是集合上的一组排列1,2,3,4,5,6. 显然，围绕中心水平轴或垂直轴有一些旋转，该轴将面号 1 带到任何其他面，因此|球体G⁡(1)|=6. 接下来，我们考虑刺G(1)。在这里，我们要求一个立方体的所有旋转都将面号 1 留在原处。当然，只有四种这样的运动——旋转0∘,90∘,180∘， 和270∘– 关于垂直于面部并通过其中心的线（见图 7.2）。因此，由定理7.4,|G|=|球体G⁡(1)||刺G⁡(1)|=6⋅4=24.

## 数学代写|抽象代数作业代写abstract algebra代考|An Application of Cosets to the Rubik’s Cube

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