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数学代写|现代代数代考Modern Algebra代写|CAYLEY’S THEOREM

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数学代写|现代代数代考Modern Algebra代写|CAYLEY’S THEOREM

We have seen that the nature of groups can vary widely-from groups of numbers to groups of permutations to groups defined by tables. Cayley’s Theorem asserts that in spite of this broad range of possibilities, each group is isomorphic to some group of permutations. This is an example of what is known as a representation theorem-it tells us that any group can be represented as (is isomorphic to, in this case) something reasonably concrete. In place of studying the given group, we can just as well study the concrete object (permutation group) representing it; and this can be an advantage. However, it can also be a disadvantage, for part of the power of abstraction comes from the fact that abstraction filters out irrelevancies, and in concentrating on any concrete object we run the risk of being distracted by irrelevancies. Still, Cayley’s Theorem has proved to be useful, and its proof ties together several of the important ideas that we have studied.

In proving Cayley’s Theorem, we associate with each element of a group $G$ a permutation of the set $G$. The way in which this is done is suggested by looking at the Cayley table for a finite group. As we observed after Theorem 14.1, each element of a finite group appears exactly once in each row of the Cayley table for the group (if we ignore the row labels at the outside of the table). Thus the elements in each row of the table are merely a permutation of the elements in the first row. What we do is simply associate with each element $a$ of $G$ the permutation whose first row (in two-row form) is the first row of the Cayley table and whose second row is the row labeled by $a$. If the elements in the first row are $a_1, a_2, \ldots, a_n$ (in that order), then the elements in the row labeled by $a$ will be $a a_1, a a_2, \ldots, a a_n$ (in that order).

Example 20.1. Consider the Cayley table for $\mathbb{Z}_6$, given in Example 11.2. The permutation associated with [3] by the idea just described is
$$\left(\begin{array}{llllll} {[0]} & {[1]} & {[2]} & {[3]} & {[4]} & {[5]} \ {[3]} & {[4]} & {[5]} & {[0]} & {[1]} & {[2]} \end{array}\right) .$$
Cayley’s Theorem extends this idea to groups that are not necessarily finite, and also establishes that this association of group elements with permutations is an isomorphism.
Cayley’s Theorem. Every group is isomorphic to a permutation group on its set of elements.

数学代写|现代代数代考Modern Algebra代写|HOMOMORPHISMS OF GROUPS. KERNELS

Definition. If $G$ is a group with operation $*$, and $H$ is a group with operation #, then a mapping $\theta: G \rightarrow H$ is a homomorphism if
$$\theta(a * b)=\theta(a) # \theta(b)$$
for all $a, b \in G$
Every isomorphism is a homomorphism. But a homomorphism need not be one-to-one, and it need not be onto.

Example 21.1. For any positive integer $n$, define $\theta: \mathbb{Z} \rightarrow \mathbb{Z}_n$ by $\theta(a)=[a]$ for each $a \in \mathbb{Z}$. Then $\theta(a+b)=[a+b]=[a] \oplus[b]=\theta(a) \oplus \theta(b)$ for all $a, b \in \mathbb{Z}$, so that $\theta$ is a homomorphism. It is onto but not one-to-one.

Example 21.2. Define $\theta: \mathbb{Z} \rightarrow \mathbb{Z}$ by $\theta(a)=2 a$ for each $a \in \mathbb{Z}$. Then $\theta(a+b)=$ $2(a+b)=2 a+2 b=\theta(a)+\theta(b)$ for all $a, b \in \mathbb{Z}$. Thus $\theta$ is a homomorphism. It is oneto-one, but not onto.

现代代数代写

数学代写|现代代数代考Modern Algebra代写|CAYLEY’S THEOREM

$$\left(\begin{array}{llllll} {[0]} & {[1]} & {[2]} & {[3]} & {[4]} & {[5]} \ {[3]} & {[4]} & {[5]} & {[0]} & {[1]} & {[2]} \end{array}\right) .$$
Cayley定理将这一思想扩展到不一定是有限的群，并且还建立了群元素与置换的这种联系是同构的。

数学代写|现代代数代考Modern Algebra代写|HOMOMORPHISMS OF GROUPS. KERNELS

$$\theta(a * b)=\theta(a) # \theta(b)$$

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MATLAB代写

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