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# 数学代写|抽象代数代写Abstract Algebra代考|Subgroups and generating sets

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## 数学代写|抽象代数代写Abstract Algebra代考|Subgroups and generating sets

As we have seen in Chapter 3 , examples of groups are quite varied: sets of numbers or matrices under addition or multiplication, sets of functions under composition, and even finite sets with a suitably constructed table are groups. However, for many of these examples, it’s not the operation that’s different, but rather only the set that changes.
Example 4.1. Consider the groups $G=\mathbb{R}$ and $H=\mathbb{Q}$. Although they are both groups in their own right, there’s a natural relationship between them: not only is $\mathbb{Q}$ a subset of $\mathbb{R}$, but their operations are identical: addition of elements of $\mathbb{Q}$ is identical to addition of those same elements in $\mathbb{R}$. When this special relationship of being both a subset and agreement on the operation occurs, we tend to think not of two different groups, but of one “large” group inside of which lies the “smaller” group as a subset.

There is a technical issue we must first raise. The binary operation of a group $\langle G, \cdot\rangle$ is a function with domain $G \times G$. Given a subset $H \subset G$, we can restrict the domain of – to $H \times H$ and see if this restricted operation – the “same” one used to define a valid operation on $G$ – yields a group (or a binary operation, at least). Hence, we need a preliminary definition to deal with this technicality.

Definition 4.2. Let $\langle G, \cdot\rangle$ be a binary structure and let $H \subset G$. Then the function $: H \times H \rightarrow G$ given by $(a, b)=a \cdot b$ is called the operation induced by $\cdot$
With this terminology, we can now define the key term precisely.
Definition 4.3. Let $G$ be a group. We say a subset $H \subset G$ is a subgroup of $G$ if and only if $H$ is a group under the operation induced by the operation on $G$. We write $H<G$ to denote that $H$ is a subgroup of $G$.

Given a group $G$, there are always two (not necessarily different) subgroups of $G$ : the group $G$ itself, and the subgroup consisting of the identity element alone. Since we’re often interested in subgroups other than those two, let’s name them for future reference.

## 数学代写|抽象代数代写Abstract Algebra代考|The center of a group

All we’ve done so far is identify what subgroups are, so we might try to use them to help us understand the structure of a group. To begin, let’s see if we can use subgroups to measure how close to abelian a given group is.

Theorem 4.13. Let $G$ be a group. Then the subset $Z(G)={g \in G \mid x g=g x$ for all $x \in G}$ is a subgroup of $G$.

Definition 4.14. Let $G$ be a group. The subgroup $Z(G)={g \in G \mid x g=g x$ for all $x \in G}$ is called the center of $G$.
Corollary 4.15. Let $G$ be a group. Then $G$ is abelian if and only if $Z(G)=G$.
Ah, an actual object that detects if a group is abelian or not. But even if a group $G$ isn’t abelian, that doesn’t mean that the center is the trivial subgroup. After all, there might be only a few elements that don’t commute. Can we use subgroups to see if an individual element commutes well?

Theorem 4.16. Let $G$ be a group and $a \in G$. Then the subset $C(a)={g \in G \mid g a=a g}$ is a subgroup of $G$.

Definition 4.17. Let $G$ be a group and $a \in G$. The subgroup $C(a)={g \in G \mid g a=a g}$ is called the centralizer of $a$ in $G$.

This means that centralizers are objects that tell us how well individual elements commute with others in the group. In fact, you might have anticipated this next theorem.
Theorem 4.18. Let $G$ be a group. Then $Z(G)=\bigcap_{a \in G} C(a)$.
Exercise 4.19. Find the centralizers of the elements $\left[\begin{array}{ll}1 & 0 \ 0 & 2\end{array}\right]$ and $\left[\begin{array}{ll}3 & 0 \ 0 & 3\end{array}\right]$ in the group $G L_2(\mathbb{R})$. Use this to state what elements are in $Z\left(G L_2(\mathbb{R})\right)$, and justify your answer.

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