Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}\)

Section6.1Preparation

ObjectivesGoals
  • Define an isomorphism of groups, guided by the desire to show two groups are "the same."
  • Verify that isomorphisms associate "same" elements in two groups.
  • Verify that isomorphisms associate "same" (sub)groups.
  • Use conjugation by an element to build isomorphisms from a group to itself ("inner automorphisms").
  • Conceive of the set of self-isomorphisms of a group (automorphisms) as a group in its own right.

Subsection6.1.1Videos

Subsection6.1.2Homework 6

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.

Subsubsection6.1.1Exercises

  1. Exercises 6.3, #3.
  2. Exercises 6.3, #4.
  3. Exercises 6.3, #5.
  4. Exercises 6.3, #7.
  5. Exercises 6.3, #8. Hint

    Note that \(\mathbb{Q}\) refers to the group of rational numbers under addition.
  6. Exercises 6.3, #12.
  7. Exercises 6.3, #28. Hint

    Use the result of Exercises 6.3 #7, together with a result from the first group exam.
  8. Exercises 6.3, #29. Hint

    It's clear why \(S_n\) is isomorphic to a subgroup of \(S_{n+2}\text{.}\) (How?) Can you make sure that the image of this isomorphism includes only even permutations?
  9. Exercises 6.3, #30. Hint

    Breadcrumbs for this proof can be found in this older video.

  10. Exercises 6.3, #34-35. Hint

    Remember, by \(\mathbb{C}\) we mean the additive group \((\mathbb{C},+)\) while by \(\mathbb{C}^*\) we mean the multiplicative group \((\mathbb{C}\setminus\{0\}, \cdot)\text{.}\)
  11. Exercises 6.3, #39.
  12. Exercises 6.3, #40.
  13. Exercises 6.3, #46.
  14. Give an example of an automorphism of \(\mathbb{Z}_6\) that is an inner automorphism induced by some element. Then, give an example of an automorphism of \(\mathbb{Z}_6\) that is an outer automorphism. Hint

    The theory in the following problem could be helpful in your search here.
  15. Let \(G\) be a group and \(a\in ZG\) be an element of its center. Prove that the inner automorphism induced by \(a\) is the identity function. Then, deduce that if \(G\) is an abelian group, then \({\rm Inn}(G)\text{,}\) the group of its inner automorphisms, is a trivial group.
  16. In \(S_4\text{,}\) consider the subgroup \begin{equation*} H = \langle (1234)\rangle = \{ (), (1234), (13)(24), (1432)\}. \end{equation*} We know that applying an inner automorphism \(\phi_\sigma\) to the subgroup \(H\) will produce another subgroup of \(S_4\text{.}\) Use this strategy to discover another subgroup of \(S_4\text{,}\) or explain why inner automorphisms can never map this particular \(H\) onto a different subgroup.