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# 数学代写|现代代数代考Modern Algebra代写|MATH310

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## 数学代写|现代代数代考Modern Algebra代写|Some Results on Finite Abelian Groups (Optional)

The aim of this section is to sample the flavor of more advanced work in groups while maintaining an acceptable level of rigor in the presentation. We attempt to achieve this balance by restricting our attention to proofs of results for abelian groups. There are instances where more general results hold, but their proofs are beyond the level of this text. In most instances of this sort, the more general results are stated informally and without proof.
The following definition of a p-group is fundamental to this entire section.

A p-group can be finite or infinite. Although we do not prove it here, a finite group is a $p$-group if and only if its order is a power of $p$. Whether or not a group is abelian has nothing at all to do with whether it is a p-group. This is brought out in the following example.

Example 1 With $p=2$, we can easily exhibit three $p$-groups of order 8 .
a. Consider first the cyclic group $C_8=\langle a\rangle$ of order 8 generated by the permutation $a=(1,2,3,4,5,6,7,8)$ :
Each of $a, a^3, a^5$, and $a^7$ has order 8 .
$a^2$ and $a^6$ have order 4 .
$a^4$ has order 2 .
The identity $e$ has order 1 .
Thus $C_8$ is a 2-group.
b. Consider now the quaternion group $G={ \pm 1, \pm i, \pm j, \pm k}$ of Exercise 34 in Section 3.1:
Each of the elements $\pm i, \pm j, \pm k$ has order 4 .
-1 has order 2 .
1 has order 1.
Hence $G$ is another 2-group of order 8 .
c. Last, consider the octic group $D_4=\left{e, \alpha, \alpha^2, \alpha^3, \beta, \gamma, \Delta, \theta\right}$ of Example 3 in Section 4.5:
Each of $\alpha$ and $\alpha^3$ has order 4.
Each of $\alpha^2, \beta, \gamma, \Delta, \theta$ has order 2 .
The identity $e$ has order 1 .
Thus $D_4$ is also a 2-group of order 8 .

## 数学代写|现代代数代考Modern Algebra代写|Cauchy’stt Theorem for Abelian Groups

If $G$ is an abelian group of order $n$ and $p$ is a prime such that $p \mid n$, then $G$ has at least one element of order $p$.

Proof The proof is by induction on the order $n$ of $G$ and uses the Second Principle of Finite Induction. For $n=1$, the theorem holds by default.

Now let $k$ be a positive integer, assume that the theorem is true for all positive integers $n<k$, and let $G$ be an abelian group of order $k$. Also, suppose that the prime $p$ is a divisor of $k$.
Consider first the case where $G$ has only the two subgroups ${0}$ and $G$. Then any $a \neq 0$ in $G$ must be a generator of $G, G=\langle a\rangle$. It follows from Exercise 38 of Section 3.4 that the order $k$ of $G$ must be a prime. Since $p$ divides this order, $p$ must equal $k$, and $G$ actually has $p-1$ elements of order $p$, by Theorem 3.26.

Now consider the case where $G$ has a nontrivial subgroup $H$; that is, $H \neq{0}$ and $H \neq G$, so that $1<|H|<k$. If $p$ divides $|H|$, then $H$ contains an element of order $p$ by the induction hypothesis, and the theorem is true for $G$. Suppose then that $p$ does not divide $|H|$. Since $G$ is abelian, $H$ is normal in $G$, and the quotient group $G / H$ has order
$$|G / H|=\frac{|G|}{|H|} .$$
We have
$$|G|=|H| \cdot|G / H|,$$

so $p$ divides the product $|H| \cdot|G / H|$. Since $p$ is a prime and $p$ does not divide $|H|, p$ must divide $|G / H|<|G|=k$. Applying the induction hypothesis, we see that the abelian group $G / H$ has an element $b+H$ of order $p$. Then
$$H=p(b+H)=p b+H,$$
and therefore $p b \in H$, where $b \notin H$. Let $r=|H|$. The order of $p b$ must be a divisor of $r$ so that $r(p b)=0$ and $p(r b)=0$. Since $p$ is a prime and $p \nmid r$, then $p$ and $r$ are relatively prime. Hence there exist integers $u$ and $v$ such that $p u+r v=1$.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Some Results on Finite Abelian Groups (Optional)

p群可以是有限的，也可以是无限的。虽然这里我们没有证明，但当且仅当一个有限群的阶是$p$的幂时，它是$p$ -群。一个群是否是阿贝尔群与它是否是p群没有任何关系。下面的例子说明了这一点。

a.首先考虑由排列$a=(1,2,3,4,5,6,7,8)$生成的8阶循环群$C_8=\langle a\rangle$:
$a, a^3, a^5$和$a^7$都是(8)阶。
$a^2$和$a^6$有顺序4。
$a^4$的阶为2。

b.现在考虑3.1节练习34的四元数群$G={ \pm 1, \pm i, \pm j, \pm k}$:

-1的阶是2。
1有o (1)

c.最后，考虑4.5节例3的octic组$D_4=\left{e, \alpha, \alpha^2, \alpha^3, \beta, \gamma, \Delta, \theta\right}$:
$\alpha$和$\alpha^3$的顺序都是4。

## 数学代写|现代代数代考Modern Algebra代写|Cauchy’stt Theorem for Abelian Groups

$$|G / H|=\frac{|G|}{|H|} .$$

$$|G|=|H| \cdot|G / H|,$$

$p$除以$|H| \cdot|G / H|$。因为$p$是质数，$p$不能除，所以$|H|, p$必须除$|G / H|<|G|=k$。应用归纳假设，我们看到阿贝尔群$G / H$有一个阶为$p$的元素$b+H$。然后
$$H=p(b+H)=p b+H,$$