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  • 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.


Subsection7.1.2Homework 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 7.5, #5 (skip part g).
  2. Exercises Section 7.5, #11.
  3. Exercises Section 7.5, #16.
  4. Exercises Section 7.5, #17.
  5. Exercises Section 7.5, #18.
  6. Exercises Section 7.5, #1.
  7. Exercises Section 7.5, #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 \(G\text{?}\)