# 数学代写|抽象代数作业代写abstract algebra代考|Math417

## 数学代写|抽象代数作业代写abstract algebra代考|External Direct Products

In this chapter, we show how to piece together groups to make larger groups. In Chapter 9 , we will show that we can often start with one large group and decompose it into a product of smaller groups in much the same way as a composite positive integer can be broken down into a product of primes. These methods will later be used to give us a simple way to construct all finite Abelian groups.
Definition External Direct Product
Let $G_1, G_2, \ldots, G_n$ be a finite collection of groups. The external direct product of $G_1, G_2, \ldots, G_n$, written as $G_1 \oplus G_2 \oplus$ $\cdots \oplus G_n$, is the set of all $n$-tuples for which the $i$ th component is an element of $G_i$ and the operation is componentwise. is an element of $G_i$ and the operation is componentwise.
In symbols,
$$G_1 \oplus G_2 \oplus \cdots \oplus G_n=\left{\left(g_1, g_2, \ldots, g_n\right) \mid g_i \in G_i\right},$$
where $\left(g_1, g_2, \ldots, g_n\right)\left(g_1^{\prime}, g_2^{\prime}, \ldots, g_n^{\prime}\right)$ is defined to be $\left(g_1 g_1^{\prime}, g_2 g_2^{\prime}, \ldots\right.$, $\left.g_n g_n^{\prime}\right)$. It is understood that each product $g_i g_i^{\prime}$ is performed with the operation of $G_i$. Note that in the case that each $G_i$ is finite, we have by properties of sets that $\left|G_1 \oplus G_2 \oplus \cdots \oplus G_n\right|=$ $\left|G_1\right|\left|G_2\right| \cdots\left|G_n\right|$. We leave it to the reader to show that the external direct product of groups is itself a group (Exercise 1).

This construction is not new to students who have had linear algebra or physics. Indeed, $\mathbf{R}^2=\mathbf{R} \oplus \mathbf{R}$ and $\mathbf{R}^3=\mathbf{R} \oplus \mathbf{R} \oplus \mathbf{R}-$ the operation being componentwise addition. Of course, there is also scalar multiplication, but we ignore this for the time being, since we are interested only in the group structure at this point.
EXAMPLE 1
\begin{aligned} U(8) \oplus U(10)=&{(1,1),(1,3),(1,7),(1,9),(3,1),(3,3),\ &(3,7)(3,9)(5,1)(5,3)(5,7)(5,9), \ &(7,1)(7,3)(7,7)(7,9)} \end{aligned}
The product $(3,7)(7,9)=(5,3)$, since the first components are combined by multiplication modulo 8 , whereas the second components are combined by multiplication modulo 10 .

## 数学代写|抽象代数作业代写abstract algebra代考|Applications

We conclude this chapter with five applications of the material presented here – three to cryptography, the science of sending and deciphering secret messages, one to genetics, and one to electric circuits.

Because computers are built from two-state electronic components, it is natural to represent information as strings of 0 s and $1 \mathrm{~s}$ called binary strings. A binary string of length $n$ can naturally be thought of as an element of $Z_2 \oplus Z_2 \oplus \cdots \oplus Z_2$ (n copies) where the parentheses and the commas have been deleted. Thus the binary string 11000110 corresponds to the element $(1,1,0,0,0,1,1,0)$ in $Z_2 \oplus Z_2 \oplus Z_2 \oplus Z_2 \oplus Z_2 \oplus Z_2 \oplus Z_2 \oplus Z_2$. Similarly, two binary strings $a_1 a_2 \cdots a_n$ and $b_1 b_2 \cdots b_n$ are added componentwise modulo 2 just as their corresponding elements in $Z_2 \oplus Z_2 \oplus \cdots \oplus Z_2$ are. For example,
$$11000111+01110110=10110001$$
and
$$10011100+10011100+00000000 .$$
The fact that the sum of two binary sequences $a_1 a_2 \cdots a_n+$ $b_1 b_2 \cdots b_n=00 \cdots 0$ if and only if the sequences are identical is the basis for a data security system used to protect Internet transactions.

Suppose that you want to purchase an item from Amazon. Need you be concerned that a hacker will intercept your creditcard number during the transaction? As you might expect, your credit-card number is sent to Amazon in a way that protects the data. We explain one way to send credit-card numbers over the Web securely. When you place an order with Amazon, the company sends your computer a randomly generated string of 0 ‘s and 1 ‘s called a key. This key has the same length as the binary string corresponding to your credit-card number and the two strings are added (think of this process as “locking” the data). The resulting sum is then transmitted to Amazon. Amazon in turn adds the string corresponding to your credit-card number (adding the key a second time “unlocks” the data).

# 抽象代数代考

## 数学代写|抽象代数作业代写abstract algebra代考|External Direct Products

$\backslash$ begin{aligned} $U(8) \backslash$ 〈oplus $U(10)=\&{(1,1),(1,3),(1,7),(1,9),(3,1),(3,3) \backslash \backslash \&(3,7)(3,9)(5,1)(5,3)(5,7)(5,9), \backslash \&(7,1)(7,3)(7,7)(7,9)} \backslash$

## 数学代写|抽象代数作业代写abstract algebra代考|Applications

$11000111+01110110=10110001$

$10011100+10011100+00000000$.

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