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Section3.2Preparation

ObjectivesLearning Goals for this Chapter
  • Determine the order of a given group, and distinguish between finite and infinite groups.
  • Determine the order of given elements in a group, and distinguish between elements of finite order and infinite order.
  • Rewrite expressions in a finite group using positive exponents only.
  • Determine whether a given subset of a group is a subgroup, using the definition and/or selecting and applying a subgroup test.
  • Determine the center of a given group, and the centralizer of a given element in a group.
  • Discover and prove relationships among the concepts of center, centralizer, and abelian groups.

The blessing (and sometime curse) of group theory is that the definition of a group provides a very minimal amount of structure: only associativity, closure, identity, and inverse properties may be taken for granted. Under this umbrella fits a surprisingly vast variety of examples of groups with very different types of elements and very different operations.

Most of our explorations of group theory in this course will focus not on groups with infinite sets of elements (such as the integers), but rather groups with finite sets of elements (such as the symmetries of a polygon). We can classify some of these finite groups as cyclic -- if all of their elements can be constructed from a single one -- and/or abelian -- if all their elements interact according to the commutative property. Where these properties do not hold in the entire group, they can hold in more limited circumstances. This leads us to look within a larger group for smaller subgroups in which properties such as cyclic and/or abelian can hold. Thinking of subgroups as potential "building blocks" for larger groups gives us a sense of why this will be a crucial concept for our later classifications of the types of finite groups that can exist.

Subsection3.2.1Acquaint 3

  1. Watch the videos above, and read Section 3.3 and leave comments, questions, and discussions using your http://hypothes.is account.
  2. Complete Acquaint 3 by 10:00am on Monday 9/24.

Subsection3.2.2Acquaint 4

  1. Watch the video above, and read Section 4.3 and leave comments, questions, and discussions using your http://hypothes.is account.
  2. Complete Acquaint 4 by 10:00am on Wednesday 9/26.

Subsection3.2.3Acquaint 5

  1. Watch the videos above, and read (if you have it) Gallian, pp. 66-68 in chapter 3.
  2. Complete Acquaint 5 by 10:00am on Friday 9/28.

Subsection3.2.4Homework 3

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

Subsubsection3.2.1Exercises

  1. Exercises 3.4, #1.
  2. Exercises 3.4, #3.
  3. Exercises 3.4, #7.
  4. Exercises 3.4, #8.
  5. Exercises 3.4, #12.
  6. Exercises 3.4, #13.
  7. Show that \(U(14) = \langle 5 \rangle\) (and hence, \(U(14)\) is cyclic).
  8. In the group \(U(40)\text{,}\) find: (a) a cyclic subgroup of order 4, and (b) a non-cyclic subgroup of order 4.
  9. Denote by \(\mathbb{R}^*\) the group of nonzero real numbers, with the operation of multiplication. Find examples of elements \(a,b \in \mathbb{R}^*\) such that all of the following hold: \begin{align*} |a|=\infty \amp\amp |b|=\infty \amp\amp \bigl|ab\bigr| = 2. \end{align*} Hint

    Begin with the last criterion. What element(s) in \(\mathbb{R}^*\) have order 2?
  10. Denote by \(\mathbb{R}^*\) the group of nonzero real numbers, with the operation of multiplication. Prove that the following subset \(H\subset \mathbb{R}^*\) is a subgroup of \(\mathbb{R}^*\text{:}\) \begin{equation*} H = \bigl\{ x \in \mathbb{R}^* \colon x^2 \text{ is rational} \bigr\}. \end{equation*} Hint

    The one-step subgroup test can make quick work of this proof.
  11. Let \(G\) be a group and \(g\in G\) be an element. Prove that if \(|g|=n\text{,}\) then \(\bigl| g^{-1} \bigr| = n\text{.}\)
  12. Let \(G\) be a group and \(a,x \in G\) be two elements. Prove that \(\bigl| x\, a\, x^{-1} \bigr| = \bigl| a \bigr|.\) Hint

    First assume that \(a\) has finite order, and let \(|a| = k\text{.}\) It's straightforward to show why the \(k\)-th power of \((xax^{-1})\) is the identity. Why can't this be true for any positive power less than \(k\text{?}\)

    You may also want to make a separate argument for the case where \(|a| = \infty\text{.}\)
  13. Let \(G\) be a group that has a unique element \(x\) such that \(|x| = 2\text{.}\) Prove that \(x \in ZG\text{.}\) Hint

    Try a contradiction proof: Assume that \(x \not\in ZG\text{.}\) This guarantees the existence of an element \(g\) such that \(xg \neq gx\text{.}\) Use this to produce a new element \(y \in G\) which is different from \(x\) as follows.

    \begin{align*} xg \amp\neq gx\\ xg(gx)^{-1} \amp\neq e\\ xgx^{-1}g^{-1} \amp\neq e \\ y = x\, xgx^{-1}g^{-1} \amp \neq x. \end{align*}

    Now, figure out the order of \(y\text{.}\)
  14. Let \(G\) be a group, and \(x,y \in G\) be two elements. Prove that \(|x\, y| = |y\, x|.\)
  15. Let \(G\) be a group, and let \(a\in G\) be an element of order \(k\text{.}\) Prove that if \(a^i = a^j\text{,}\) then \(i-j\) must be a multiple of \(k\text{.}\)
  16. Let \(G\) be a finite group of order \(n\text{,}\) and let \(a\in G\) be an element of order \(k\text{.}\) Prove that \(k \leq n\text{.}\) (In other words, the order of any element in a finite group cannot be larger than the order of the group.) Hint

    Use the result of the previous exercise.
  17. Let \(G\) be an abelian group. Show that the following subset \(H\) is a subgroup of \(G\text{:}\) \begin{equation*} H = \bigl\{ x \in G \colon |x| \text{ is odd} \bigr\} \end{equation*} Hint

    Remember that we're assuming \(G\) is abelian...! This should make it possible to use the one-step subgroup test. If \(|x| = k\) is odd and \(|y| = \ell\) is odd, can you say with certainty what \(\bigl| xy^{-1} \bigr|\) is? How many powers will make it into the identity?
  18. Let \(G\) be an abelian group. Show that the following subset \(T\) is a subgroup of \(G\text{:}\) \begin{equation*} T = \bigl\{ x \in G \colon |x| \text{ is finite} \bigr\} \end{equation*} This subgroup is called the torsion subgroup of \(G\text{.}\) Hint

    As in the previous exercise, try the one-step subgroup test and remember that we're assuming \(G\) is abelian.
  19. Let \(G\) be a group and \(a\in G\) be an element. Prove that \(C(a) = C\bigl(a^{-1}\bigr)\text{.}\) Hint

    When you restate this equality of sets using an element argument, you get the logical proposition "for all \(x\in G\text{,}\) \(x\) commutes with \(a\) if and only if \(x\) commutes with \(a^{-1}\text{.}\)"
  20. Let \(G\) be a group and \(a\in G\) be an element. Prove that for any integer \(k\text{,}\) we have \begin{equation*} C(a) \subset C\bigl(a^k\bigr). \end{equation*} Hint

    The case \(k=0\) should be quick. You might then prove the statement assuming \(k \geq 1\text{,}\) and finally use the previous exercise to show why the negative values of \(k\) follow immediately.