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# 数学代写|抽象代数代写Abstract Algebra代考|MATH4200 Classification of Groups of Order Up to 15

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## 数学代写|抽象代数代写Abstract Algebra代考|Classification of Groups of Order Up to 15

The next theorem illustrates the utility of the ideas presented in this chapter.
Theorem 25.4 Classification of Groups of Order 8 (Cayley, 1859)
Up to isomorphism, there are only five groups of order 8 :
$Z_8, Z_4 \oplus Z_2, Z_2 \oplus Z_2 \oplus Z_2, D_4$, and the quaternions.

PROOF The Fundamental Theorem of Finite Abelian Groups takes care of the Abelian cases. Now, let $G$ be a non-Abelian group of order 8 . Also, let $G_1=\left\langle a, b \mid a^4=b^2=(a b)^2=e\right\rangle$ and let $G_2=\left\langle a, b \mid a^2=b^2=(a b)^2\right\rangle$. We know from the preceding examples that $G_1$ is isomorphic to $D_4$ and $G_2$ is isomorphic to the quaternions. Thus, it suffices to show that $G$ must satisfy the defining relations for $G_1$ or $G_2$. It follows from Exercise 45 in Chapter 2 and Lagrange’s Theorem that $G$ has an element of order 4 ; call it $a$. Then, if $b$ is any element of $G$ not in $\langle a\rangle$, we know that
$$G=\langle a\rangle \cup\langle a\rangle b=\left{e, a, a^2, a^3, b, a b, a^2 b, a^3 b\right} .$$

## 数学代写|抽象代数代写Abstract Algebra代考|Characterization of Dihedral Groups

As another nice application of generators and relations, we will now give a characterization of the dihedral groups that has been known for more than 100 years. For $n \geq 3$, we have used $D_n$ to denote the group of symmetries of a regular $n$-gon. Imitating Example 2, one can show that
$D_n \approx\left\langle a, b \mid a^n=b^2=(a b)^2=e\right\rangle$ (see Exercise 9). By analogy, these generators and relations serve to define $D_1$ and $D_2$ also. (These are also called dihedral groups.) Finally, we define the infinite dihedral group $D_{\infty}$ as $\left\langle a, b \mid a^2=b^2=e\right\rangle$. The elements of $D_{\infty}$ can be listed as $e, a, b, a b, b a,(a b) a,(b a) b,(a b)^2,(b a)^2,(a b)^2 a,(b a)^2 b,(a b)^3,(b a)^3, \ldots$
Theorem 25.5. Characterization of Dihedral Groups
Any group generated by a pair of elements of order 2 is dihedral.
PROOF Let $G$ be a group generated by a pair of distinct elements of order 2 , say, $a$ and $b$. We consider the order of $a b$. If $|a b|=\infty$, then $G$ is infinite and satisfies the relations of $D_{\infty}$. We will show that $G$ is isomorphic to $D_{\infty}$. By Dyck’s Theorem, $G$ is isomorphic to some factor group of $D_{\infty}$, say, $D_{\infty} / H$. Now, suppose $h \in H$ and $h \neq e$. Since every element of $D_{\infty}$ has one of the forms $(a b)^i,(b a)^i,(a b)^i a$, or $(b a)^i b$, by symmetry, we may assume that $h=(a b)^i$ or $h=(a b)^i a$. If $h=(a b)^i$, we will show that $D_{\infty} / H$ satisfies the relations for $D_i$ given in Exercise 9. Since $(a b)^i$ is in $H$, we have
$$H=(a b)^i H=(a b H)^i$$

so that $(a b H)^{-1}=(a b H)^{i-1}$. But
$$(a b)^{-1} H=b^{-1} a^{-1} H=b a H,$$
and it follows that
$$a H a b H a H=a^2 H b H a H=e H b a H=b a H=(a b H)^{-1} .$$

## 数学代写|抽象代数代写Abstract Algebra代考|Classification of Groups of Order Up to 15

$Z_8, Z_4 \oplus Z_2, Z_2 \oplus Z_2 \oplus Z_2, D_4$ 和四元数。

left 缺少或无法识别的分隔符

## 数学代写|抽象代数代写Abstract Algebra代考|Characterization of Dihedral Groups

$D_n \approx\left\langle a, b \mid a^n=b^2=(a b)^2=e\right\rangle$ (见练习9) 。以此类推，这些生成器和关系用于定义 $D_1$ 和 $D_2$ 还。 (这些也称为二面角群。) 最后，我们定义无限二面角群 $D_{\infty}$ 作为 $\left\langle a, b \mid a^2=b^2=e\right\rangle$. 的 元素 $D_{\infty}$ 可以列为 $e, a, b, a b, b a,(a b) a,(b a) b,(a b)^2,(b a)^2,(a b)^2 a,(b a)^2 b,(a b)^3,(b a)^3, \ldots$ 定理 25.5。二面角群的特征

$(a b)^i,(b a)^i,(a b)^i a$ ，或者 $(b a)^i b$ ，通过对称性，我们可以假设 $h=(a b)^i$ 或者 $h=(a b)^i a$. 如果
$$H=(a b)^i H=(a b H)^i$$

$$(a b)^{-1} H=b^{-1} a^{-1} H=b a H,$$

$$a H a b H a H=a^2 H b H a H=e H b a H=b a H=(a b H)^{-1} .$$

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