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# 数学代写|现代代数代考Modern Algebra代写|MATG5020 Groups and subgroups

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## 数学代写|现代代数代考Modern Algebra代写|Definition and basic properties of groups

We’ll look at basic properties of groups, and since we’ll discuss groups in general, we’ll use a multiplicative notation even though some of the example groups are Abelian.

Definition 4.1. The axioms for a group are very few. A group $G$ has an underlying set, also denoted $G$, and a binary operation $G \times G \rightarrow G$ that satisfies three properties.

Associativity. $(x y) z=x(y z)$.

Identity. There is an element 1 such that $1 x=x=x 1$.

Inverses. For each element $x$ there is an element $x^{-1}$ such that $x x^{-1}=x^{-1} x=1$.
Theorem 4.2. From these few axioms several properties of groups immediately follow.

Uniqueness of the identity. There is only one element $e$ such that $e x=x=x e$, and it is $e=1$.

Outline of proof. The definition says that there is at least one such element. To show that it’s the only one, suppose $e$ also has the property of an identity and prove $e=1$.

Uniqueness of inverses. For each element $x$ there is only one element $y$ such that $x y=y x=1$.

Outline of proof. The definition says that there is at least one such element. To show that it’s the only one, suppose that $y$ also has the property of an inverse of $x$ and prove $y=x^{-1}$.

Inverse of an inverse. $\left(x^{-1}\right)^{-1}=x$.
Outline of proof. Show that $x$ has the property of an inverse of $x^{-1}$ and use the previous result.

Inverse of a product. $(x y)-1=y^{-1} x^{-1}$.
Outline of proof. Show that $y^{-1} x^{-1}$ has the property of an inverse of $x y$.

Cancellation. If $x y=x z$, then $y=z$, and if $x z=y z$, then $x=y$.

Solutions to equations. Given elements $a$ and $b$ there are unique solutions to each of the equations $a x=b$ and $y a=b$, namely, $x=a^{-1} b$ and $y=b a^{-1}$.

Generalized associativity. The value of a product $x_1 x_2 \cdots x_n$ is not affected by the placement of parentheses.

Outline of proof. The associativity in the definition of groups is for $n=3$. Induction is needed for $n>3$.

Powers of an element. You can define $x^n$ for nonnegative values of $n$ inductively. For the base case, define $x^0=1$, and for the inductive step, define $x^{n+1}=x x^n$. For negative values of $n$, define $x^n=\left(x^{-n}\right)^{-1}$.

Properties of powers. Using the definition above, you can prove using induction the following properties of powers where $m$ and $n$ are any integers: $x^m x^n=x^{m+n},\left(x^m\right)^n=$ $x^{m n}$.
Note that $(x y)^n$ does not equal $x^n y^n$ in general, although it does for Abelian groups.

## 数学代写|现代代数代考Modern Algebra代写|Subgroups

A subgroup $H$ of $G$ is a group whose underlying set is a subset of the underlying set of $G$ and has the same binary operation, that is, for $x, y \in H, x \cdot_H y=x \cdot_G y$ where $\cdot_H$ denotes is the multiplication in $H$ while $\cdot_G$ denotes is the multiplication in $G$. Since they are the same, we won’t have to subscript the multiplication operation.

An alternate description of a subgroup $H$ is that it is a subset of $G$ that is closed under multiplication, has 1 , and is closed under inverses.

Of course, $G$ is a subgroup of itself. All other subgroups of $G$, that is, those subgroups that don’t have every element of $G$ in them, are called proper subgroups.

Also, ${1}$ is a subgroup of $G$, usually simply denoted 1 . It’s called the trivial subgroup of $G$.

Example 4.3. Consider the cyclic group of six elements $G=\left{1, a, a^2, a^3, a^4, a^5\right}$ where $a^6=1$. Besides the trivial subgroup 1 and the entire subgroup $G$, there are two other subgroups of $G$. One is the 3-element subgroup $\left{1, a^2, a^4\right}$ and the other is the 2-element $\operatorname{subgroup}\left{1, a^3\right}$.

The intersection $H \cap K$ of two subgroups $H$ and $K$ is also a subgroup, as you can easily show. Indeed, the intersection of any number of subgroups is a subgroup.

The union of two subgroups is never a subgroup unless one of the two subgroups is contained in the other.
Exercise 48. About intersections and unions of subgroups.
(a). Show that the intersection of two subgroups is also a subgroup.
(b). Give a counterexample where the union of two subgroups is not a subgroup.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Subgroups

$x, y \in H, x \cdot H y=x \cdot G$ 在哪里·$\cdot H$ 表示是乘法 $H$ 尽管· $G$ 表示是乘法 $G$. 因为它们是相同的，所以我们不必为 乘法运算加上下标。

(A)。证明两个子群的交集也是一个子群。
(二). 给出一个反例，其中两个子群的并集不是子群。

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