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

Section6.2Preparation

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.

Subsection6.2.2Homework 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.

Subsubsection6.2.1Exercises

1. Exercises 6.4, #3.
2. Exercises 6.4, #4.
3. Exercises 6.4, #5.
4. Exercises 6.4, #7.
5. Exercises 6.4, #8. Hint

Note that $\mathbb{Q}$ refers to the group of rational numbers under addition.

6. Exercises 6.4, #12.
7. Exercises 6.4, #28. Hint

Use the result of Exercises 6.4 #7, together with a result from the first group exam.

8. Exercises 6.4, #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 6.4, #30. Hint

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

10. Exercises 6.4, #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 6.4, #39.
12. Exercises 6.4, #40.
13. Exercises 6.4, #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.