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## Section7.2Preparation

###### ObjectivesGoals
• Compare and contrast the effects of left- and right-actions, and inner automorphisms, on a subgroup $H \lt G\text{.}$
• Given a subgroup $H \lt G\text{,}$ define the cosets of $H$ in $G$ using left- and right-actions induced by elements of $G\text{.}$
• Define cosets of $H$ using an equivalence relation on the elements of $G\text{,}$ and thereby prove that the cosets of $H$ partition $G\text{.}$
• Deduce Lagrange's Theorem for finite groups from the above, and some of its consequences.

### Subsection7.2.2Homework 7

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.

1. Exercises Section 7.6, #5 (skip part g).
2. Exercises Section 7.6, #11.
3. Exercises Section 7.6, #16.
4. Exercises Section 7.6, #17.
5. Exercises Section 7.6, #18.
6. Exercises Section 7.6, #1.
7. Exercises Section 7.6, #9.
8. Let $G$ be a group whose order is $pq\text{,}$ where $p,q$ are distinct primes. Prove that if $H\lt G$ is a proper subgroup, then $H$ must be cyclic.
9. Let $G$ be an abelian group with an odd number of elements. Prove that $\prod_{g\in G} g = e\text{.}$ (In other words, the product of all the elements in $G$ is the identity.)
10. Let $|G|=14\text{,}$ and assume $G$ has only one subgroup of order 2 and one subgroup of order 7. Prove that $G$ must be cyclic.
11. Let $|G|=63\text{.}$ Prove that $G$ must have an element of order 3.
12. Let $G$ be a group that has elements of order $k$ for all $1\leq k \leq 12\text{.}$ What's the smallest possible order for the group $G\text{?}$