Math 441 - Abstract Algebra - Schedule

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Just as we spanned sets of permutations, we can span subsets of a group. After defining this, we'll show that spanning a subset of a group always gets you a subgroup of the group. This should be analogous to the fact that spanning a subset of permutations leads to a closed set of permutations.

Definition (The Subgroup Generated By A Subset)

Let $G$ be a group. Suppose that $S$ is a subset of $G$.

A word from the alphabet $S$ is a product of the form $$x=s_1s_2\cdots s_k$$ where for each $i$ either $s_i\in S$ or $s_i^{-1}\in S$.
The subgroup generated by $S$ is the set of words from the alphabet $S$, namely $$\left<S\right> = \{s_1s_2\cdots s_k\mid k\in \mathbb{N} \text{ and either $s_i\in S$ or $s_i^{-1}\in S$ for each } i\in\{1,2,\ldots, k\}\}.$$
If $H=\left<S\right>$, then we say that $H$ is the subgroup generated by $S$, or that $S$ is a generating set for $H$.
If $S=\{a\}$, then we'll write $\left<a\right>$ instead of $\left<\{a\}\right>$. We call $\left<a\right>$ the subgroup generated by $a$.

We'll be using this notation to develop the game of scoring for groups. We'll call the game "Generate/Don't Generate."

We call $\left<S\right>$ the subgroup of $G$ generated by $S$, but we have not yet shown this is in fact a subgroup. The next problem has you verify this, so that we can definitely refer to $\langle S\rangle$ as a subgroup.

Problem 51 (The Subgroup Generated By S Is Actually A Subgroup)

Let $G$ be a group and $S$ be a nonempty subset of $G$. Prove that $\left<S\right>$ is a subgroup of $G$.

Any time you want to show something is a subgroup, the subgroup test (see problem (Subgroups Are Subsets That Are Closed Under Products And Inverses)) will simplify your work. So you just need to show that $\left<S\right>$ is a nonempty subset of $G$ and if $a,b\in \left<S\right>$, then both $ab\in \left<S\right>$ and $a^{-1}\in \left<S\right>$.

Definition ($|a|$ and $|G|$ - Order For Elements and Groups)

Let $G$ be a group with identity $e$, and let $g\in G$.

The $\textdef{order}$ of $G$, denoted $|G|$, is the cardinality of $G$.
The $\textdef{order}$ of $g$, denoted $|g|$, is the smallest positive integer $n$ such that $g^n = e$, if such an $n$ exists. If no such $n$ exists, we say $g$ has infinite order.

Definition (The Euler Phi Function)

The Euler phi function $\varphi:\mathbb{Z}\to\mathbb{Z}$ is defined by letting $\varphi(n)$ equal the order of $U(n)$.

A graph of the first 1000 values of $\varphi(n)$. See Wikipedia.

Problem 48 (Orders Of $\mathbb{Z}_n$ And $U(n)$ And Their Elements)

We've already shown that $\mathbb{Z}_n$ under addition mod $n$ and $U(n)$ under multiplication mod $n$ are groups.

For each $n$ between 2 and 10, compute the order of $Z_n$ and the order of each element of $\mathbb{Z}_n$. Organize your work into a table where you first make a list of the elements, and then underneath each element state the order. For example, if $n=6$ then our table would look like the one below. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \text{Element}&0&1&2&3&4&5\\\hline \text{Order}&1&6&3&2&3&6\\\hline \end{array}$$
For each $n$ between 2 and 10, compute the order of $U(n)$ (state $\varphi(n)$) and then compute the order of each element of $U(n)$. You can check your work with the sage code below. I kept the numbers under 10 because you can do these by hand (or in your head) fairly quickly. Make sure you do enough by hand that you feel comfortable with this process.
You should have noticed that $U(8)$ and $U(10)$ both have order 4. Is there a 1-1 correspondence between these two groups that matches elements with the same order?

Here's some Sage code you can use to check your computations with $U(n)$.

for n in (2..20):
 Zn = Integers(n)
 Un = [x for x in Zn if gcd(ZZ(x), n) == 1]  #This creates Un
 show(table(["U("+str(n)+r") has order $\varphi($"+str(n)+"$)=$"+str(len(Un))+
             ".The elements, with orders below them, are listed below."]))
 orders=[x.multiplicative_order() for x in Un] #This computes the multiplicative order of each element.
 show(table([Un,orders])) #This creates a table of elements (top row) and their orders (bottom row).

Problem 51 (Inverses In Groups)

Suppose that $G$ is a group with $a,b\in G$.

Prove that the inverse of $a^{-1}$ is $a$.
Prove that the inverse of $ab$ is $b^{-1}a^{-1}$.
If $a_1,a_2,a_3,\ldots, a_n\in G$, state the inverse of $a_1a_2a_3\cdots a_n$. Use induction to prove your claim.

If you see yourself repeating an induction proof similar to what we've been doing, then you're on the right track.

Problem 53 (Finite Subgroup Test)

Let $G$ be a group. Suppose that $H$ is a nonempty finite subset of $G$ and that $H$ is closed under the operation of $G$ (so if $a,b\in H$, then we must have $ab\in H$). Prove that $H$ is a subgroup of $G$.

Hint. If you use the subgroup test (show that $H$ is a nonempty subset of $G$ that is closed under the operation and taking inverses), then we get to assume it's a nonempty subset of $G$ that is closed under the operation. All you have to do is explain why it's closed under taking inverses. The work you did in the problem The Inverse In A Finite Group Is A Power Of The Element should help you quite a bit. However, you can't use this theorem directly because you do not know that $G$ is a finite group. You'll want to reuse your work from that problem, not refer to the problem.

Definition (Abelian Group)

Let $G$ be a group. If $ab=ba$ for every $a,b\in G$ (so the group operation is commutative), then we say that $G$ is Abelian.

Definition ($Z(G)$ - Center Of A Group)

Let $G$ be a group. The center of the group, written $Z(G)$, is the set of elements $x\in G$ that commute with every element of $G$, which we can write symbolically as $$Z(G)=\{x\in G\mid gx=xg \text{ for all } g\in G\}.$$

The $Z$ comes from the german word "Zentrum" (see Wikipedia).

Problem 56 (The Center Of Group Is A Subgroup)

Prove that the center $Z(G)$ of a group $G$ is a subgroup of $G$. If $G$ is Abelian, then what is $Z(G)$?

For more problems, see AllProblems

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