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数学代写|抽象代数代写abstract algebra代考|algebraic properties
In studying groups we are interested in their algebraic properties, and not in the particular form in which they are presented. For example, if we construct the multiplication tables for two finite groups and find that they have the same patterns, although the elements might have different forms, then we would say that the groups have exactly the same algebraic properties.
Consider the subgroup ${\pm 1}$ of $\mathbf{Q}^{\times}$and the group $\mathbf{Z}{2}$. If you write out the group tables for these groups you will find precisely the same pattern, as shown in Tables 3.4.1 and 3.4.2. Table 3.4.1: Multiplication in ${\pm 1}$ \begin{tabular}{r|rr} $\times$ & 1 & $-1$ \ \hline 1 & 1 & $-1$ \ $-1$ & $-1$ & 1 \end{tabular} Table 3.4.2: Addition in $\mathbf{Z}{2}$
\begin{tabular}{c|cc}
$+$ & {$[0]$} & {$[1]$} \
\hline$[0]$ & {$[0]$} & {$[1]$} \
{$[1]$} & {$[1]$} & {$[0]$}
\end{tabular}
Actually, if we have any group $G$ with two elements, say the identity element $e$ and one other element $a$, then there is only one possibility for the multiplication table for $G$. We have already observed that Propositions 3.1.7 and 3.1.8 imply that in each row and column of a group table, each element of the group must occur exactly once. Since $e$ is the identity element, $e \cdot e=e, e \cdot a=a$, and $a \cdot e=a$. Since $a$ cannot be repeated in the last row of the table, we must have $a \cdot a=e$.
数学代写|抽象代数代写abstract algebra代考|isomorphic
We need a formal definition to describe when two groups have the same algebraic properties. To begin with, there should be a one-to-one correspondence between the elements of the groups. This means in essence that elements of one group could be renamed to correspond exactly to the elements of the second group. Furthermore, products of corresponding elements should correspond. If $G_{1}$ is a group with operation $*$ and $G_{2}$ is a group with operation $\star$, then any function $\phi: G_{1} \rightarrow G_{2}$ that preserves products must have the property that $\phi(a * b)=\phi(a) \star \phi(b)$ for all $a, b \in G_{1}$. This expresses in a formula the fact that if we first multiply $a$ and $b$ to exactly the same answer as if we find the corresponding elements $\phi(a)$ and $\phi(b)$ in exactly the same answer as if we find the corresponding elements $\phi(a)$ and $\phi(b)$ in $G_{2}$ and then compute their product $\phi(a) \star \phi(b)$ in $G_{2}$. It is rather cumbersome to write the two operations, so in our definition we will omit them, since it should be clear from the context which operation is to be used. A one-to-one correspondence words isos meaning “equal” and morphe meaning “form.”
3.4.1 Definition. Let $G_{1}$ and $G_{2}$ be groups, and let $\phi: G_{1} \rightarrow G_{2}$ be a function. Then $\phi$ is said to be a group isomorphism if $\phi$ is one-to-one and onto and
$$
\phi(a b)=\phi(a) \phi(b)
$$
for all $a, b \in G_{1}$. In this case, $G_{1}$ is said to be isomorphic to $G_{2}$, and this is denoted by $G_{1} \cong G_{2}$.
抽象代数代写
数学代写|抽象代数代写ABSTRACT ALGEBRA代考|ALGEBRAIC PROPERTIES
在研究小组中，我们对它们的代数性质感兴趣，而不是对它们呈现的特定形式感兴趣。例如，如果我们为两个有限群构建乘法表并发现它们具有相同的模式，尽管元素可能具有不同的形式，那么我们可以说这些群具有完全相同的代数性质。
考虑子 ${\pm 1}$ of $\mathbf{Q}^{\times}$and the group $\mathbf{Z}{2}$. If you write out the group tables for these groups you will find precisely the same pattern, as shown in Tables 3.4.1 and 3.4.2. Table 3.4.1: Multiplication in ${\pm 1}$ \begin{tabular}{r|rr} $\times$ & 1 & $-1$ \ \hline 1 & 1 & $-1$ \ $-1$ & $-1$ & 1 \end{tabular} Table 3.4.2: Addition in $\mathbf{Z}{2}$
\begin{tabular}{c|cc}
$+$ & {$[0]$} & {$[1]$} \
\hline$[0]$ & {$[0]$} & {$[1]$} \
{$[1]$} & {$[1]$} & {$[0]$}
\end{tabular}
Actually, if we have any group $G$ with two elements, say the identity element $e$ and one other element $a$, then there is only one possibility for the multiplication table for $G$. We have already observed that Propositions 3.1.7 and 3.1.8 imply that in each row and column of a group table, each element of the group must occur exactly once. Since $e$ is the identity element, $e \cdot e=e, e \cdot a=a$, and $a \cdot e=a$. Since $a$ cannot be repeated in the last row of the table, we must have $a \cdot a=e$.
数学代写|抽象代数代写ABSTRACT ALGEBRA代考|ISOMORPHIC
我们需要一个正式的定义来描述两个群何时具有相同的代数性质。首先，组的元素之间应该存在一一对应的关系。这实质上意味着一个组的元素可以重命名为与第二组的元素完全对应。此外，相应元素的产品应该对应。如果G1是一个有操作的组∗和G2是一个有操作的组⋆, 那么任何函数φ:G1→G2保存产品必须具有以下属性φ(一种∗b)=φ(一种)⋆φ(b)对全部一种,b∈G1. 这在一个公式中表达了这样一个事实，即如果我们先相乘一种和b得到完全相同的答案，就像我们找到相应的元素一样φ(一种)和φ(b)在完全相同的答案中，就好像我们找到了相应的元素φ(一种)和φ(b)在G2然后计算他们的产品φ(一种)⋆φ(b)在G2. 编写这两个操作相当麻烦，因此在我们的定义中我们将省略它们，因为从上下文中应该清楚要使用哪个操作。一对一的对应词 isos 意思是“相等”，morphe 意思是“形式”。
3.4.1 定义。让G1和G2成为组，并让φ:G1→G2成为一个函数。然后φ被称为群同构如果φ是一对一的，并且
φ(一种b)=φ(一种)φ(b)
对全部一种,b∈G1. 在这种情况下，G1据说是同构的G2，这表示为G1≅G2.
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