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# 数学代写|现代代数代考Modern Algebra代写|MATH612

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## 数学代写|现代代数代考Modern Algebra代写|Isomorphisms

It turns out that the permutation groups can serve as models for all groups. For this reason, we examine permutation groups in great detail in the next chapter. In order to describe their relation to groups in general, we need the concept of an isomorphism. Before formally introducing this concept, however, we consider some examples.

Example 1 Consider a cyclic group of order 4. If $G$ is a cyclic group of order 4, it must contain an identity element $e$ and a generator $a \neq e$ in $G$. The proof of Theorem 3.21 shows that
$$G=\left{e, a, a^2, a^3\right}$$
where $a^4=e$. A multiplication table for $G$ would have the form shown in Figure 3.14.

In a very definite way, then, the structure of $G$ is determined. The details as to what the element $a$ might be and what the operation in $G$ might be may vary, but the basic structure of $G$ fits the pattern in the table.

Example 2 Let us consider a group related to geometry. We begin with an equilateral triangle $T$ with center point $O$ and vertices labeled $V_1, V_2$, and $V_3$ (see Figure 3.15).

The equilateral triangle, of course, consists of the set of all points on the three sides of the triangle. By a rigid motion of the triangle, we mean a bijection of the set of points of the triangle onto itself that leaves the distance between any two points unchanged. In other words, a rigid motion of the triangle is a bijection that preserves distances. Such a rigid motion must map a vertex onto a vertex, and the entire mapping is determined by the images of the vertices $V_1, V_2$, and $V_3$. These rigid motions (or symmetries, as they are often called) form a group with respect to mapping composition. (Verify this.) There are a total of six elements in the group, and they may be described as follows:

1. $e$, the identity mapping, that leaves all points unchanged.
2. $r$, a counterclockwise rotation through $120^{\circ}$ about $O$ in the plane of the triangle.
3. $r^2=r \circ r$, a counterclockwise rotation through $240^{\circ}$ about $O$ in the plane of the triangle.
4. A reflection $f$ about the line $L_1$ through $V_1$ and $O$.
5. A reflection $g$ about the line $L_2$ through $V_2$ and $O$.
6. A reflection $h$ about the line $L_3$ through $V_3$ and $O$.

## 数学代写|现代代数代考Modern Algebra代写|Homomorphisms

We saw in the last section that an isomorphism between two groups provides a connection that shows that the two groups have the same structure relative to their group operations. It is for this reason that the concept of an isomorphism is extremely important in algebra.
The name homomorphism is given to another important type of mapping that is related to, but different from, the isomorphism. The basic differences are that a homomorphism is not required to be one-to-one and also not required to be onto. The formal definition is as follows.
Homomorphism, Endomorphism, Epimorphism, Monomorphism phism from $G$ to $G^{\prime}$ is a mapping $\phi: G \rightarrow G^{\prime}$ such that
$$\phi(x \circledast y)=\phi(x) \text { 函 } \phi(y)$$
for all $x$ and $y$ in $G$. If $G=G^{\prime}$, the homomorphism $\phi$ is an endomorphism. A homomorphism $\phi$ is called an epimorphism if $\phi$ is onto, and a monomorphism if $\phi$ is one-to-one.
If there exists an epimorphism from $G$ to $G^{\prime}$, then $G^{\prime}$ is called a homomorphic image of $G$.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Isomorphisms

$$G=\left{e, a, a^2, a^3\right}$$

$e$即恒等映射，使所有点保持不变。

$r$，在三角形平面上绕$O$逆时针旋转$120^{\circ}$。

$r^2=r \circ r$，在三角形平面上绕$O$逆时针旋转$240^{\circ}$。

## 数学代写|现代代数代考Modern Algebra代写|Homomorphisms

$$\phi(x \circledast y)=\phi(x) \text { 函 } \phi(y)$$

## MATLAB代写

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