# 数学代写|现代代数代写Modern Algebra代考|MATH612

#### Doug I. Jones

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## 数学代写|现代代数代写Modern Algebra代考|Cayley’s Theorem

Every group is isomorphic to a group of permutations.
Proof Let $G$ be a given group. The permutations that we use in the proof will be mappings defined on the set of all elements in $G$.
For each element $a$ in $G$, we define a mapping $f_a: G \rightarrow G$ by
$$f_a(x)=a x \text { for all } x \text { in } G .$$

That is, the image of each $x$ in $G$ is obtained by multiplying $x$ on the left by $a$. Now, $f_a$ is one-to-one since
\begin{aligned} f_a(x)=f_a(y) & \Rightarrow a x=a y \ & \Rightarrow \quad x=y . \end{aligned}
To see that $f_a$ is onto, let $b$ be arbitrary in $G$. Then $x=a^{-1} b$ is in $G$, and for this particular $x$ we have
\begin{aligned} f_a(x) & =a x \ & =a\left(a^{-1} b\right)=b . \end{aligned}
Thus $f_a$ is a permutation on the set of elements of $G$.
We shall show that the set
$$G^{\prime}=\left{f_a \mid a \in G\right}$$
actually forms a group of permutations. Since mapping composition is always associative, we only need to show that $G^{\prime}$ is closed, has an identity, and contains inverses.
For any $f_a$ and $f_b$ in $G^{\prime}$, we have
$$f_a f_b(x)=f_a\left(f_b(x)\right)=f_a(b x)=a(b x)=(a b)(x)=f_{a b}(x)$$
for all $x$ in $G$. Thus $f_a f_b=f_{a b}$, and $G^{\prime}$ is closed. Since
$$f_e(x)=e x=x$$
for all $x$ in $G, f_e$ is the identity permutation, $f_e=I_G$. Using the result $f_a f_b=f_{a b}$, we have
$$f_a f_{a^{-1}}=f_{a a^{-1}}=f_e$$
and
$$f_{a^{-1}} f_a=f_{a^{-1} a}=f_e .$$

## 数学代写|现代代数代写Modern Algebra代考|Permutation Groups in Science and Art (Optional)

Often, the usefulness of some particular knowledge in mathematics is neither obvious nor simple. So it is with permutation groups. Their applications in the real world come about through connections that are somewhat involved. Nevertheless, we shall indicate here some of their uses in both science and art.

Most of the scientific applications of permutation groups are in physics and chemistry. One of the most impressive applications occurred in 1962. In that year, physicists Murray Gell-Mann and Yuval Ne’eman used group theory to predict the existence of a new particle, which was designated the omega minus particle. It was not until 1964 that the existence of this particle was confirmed in laboratory experiments.

One of the most extensive uses made of permutation groups has been in the science of crystallography. As mentioned in Section 4.1, every geometric figure in two or three dimensions has its associated rigid motions, or symmetries. This association provides a natural connection between permutation groups and many objects in the real world. One of the most fruitful of these connections has been made in the study of the structure of crystals. Crystals are classified according to geometric symmetry based on a structure with a balanced arrangement of faces. One of the simplest and most common examples of such a structure is provided by the fact that a common table salt $(\mathrm{NaCl})$ crystal is in the shape of a cube.

In this section, we examine some groups related to the rigid motions of a plane figure. We have already seen two examples of this type of group. The first was the dihedral group $D_3$, the group of symmetries of an equilateral triangle in Example 2 of Section 3.5, and the other dihedral group was the octic group $D_4$, the group of symmetries of a square in Example 12 of Section 4.1.

It is not hard to see that the symmetries of any plane figure $F$ form a group under mapping composition. We already know that the permutations on the set $F$ form a group $\mathcal{S}(F)$ with respect to mapping composition. The identity permutation $I_F$ preserves distances and consequently is a symmetry of $F$. If two permutations on $F$ preserve distances, their composition does also, and if a given permutation preserves distances, its inverse does also. Thus the symmetries of $F$ form a subgroup of $\mathcal{S}(F)$.

# 现代代数代考

## 数学代写|现代代数代写Modern Algebra代考|Cayley’s Theorem

$$f_a(x)=a x \text { for all } x \text { in } G .$$

\begin{aligned} f_a(x)=f_a(y) & \Rightarrow a x=a y \ & \Rightarrow \quad x=y . \end{aligned}

\begin{aligned} f_a(x) & =a x \ & =a\left(a^{-1} b\right)=b . \end{aligned}

$$G^{\prime}=\left{f_a \mid a \in G\right}$$

$$f_a f_b(x)=f_a\left(f_b(x)\right)=f_a(b x)=a(b x)=(a b)(x)=f_{a b}(x)$$

$$f_e(x)=e x=x$$

$$f_a f_{a^{-1}}=f_{a a^{-1}}=f_e$$

$$f_{a^{-1}} f_a=f_{a^{-1} a}=f_e .$$

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