Lecture 10

Navigation

• Back to Main Page.
• Back to the Lectures.
• Before the Lecture
• Related Problems
• In Class
• Outcomes

Before the Lecture

• Read Section 2.3:
  • Definition and basic properties of Groups.
  • Examples (Example 2.47, except (v), (vii) and (viii), and 2.48, except (vi)).
  • Associativity and parentheses (Theorem 2.49)
  • Powers/multiplication by integers.
  • Order of elements (Definition and Propositions 2.54 and 2.55).
  • Skip from Symmetry (pg. 137) to the end of the section.
• Watch the videos related to this section (after reading it):
  • Remarks on abstract groups and examples.
  • Powers and multiplicative/additive notations.
  • Order of elements.
  • Problems 2.36(v), 2.40.
• Read Section 2.4:
  • Definition of subgroups.
  • Subgroup criteria (Propositions 2.68 and 2.69).
  • Alternating subgroups \(A_n\).
  • Cyclic subgroups.
  • Skip from Proposition 2.71 to 2.73.
  • Order of a group.
  • Skip from Proposition 2.75 to Lemma 2.82.
  • Lagrange's Theorem (Theorem 2.83 -- you can skip the proof) and its corollaries (from 2.85 to 2.87, you can skip 2.84).
• Watch the videos related to this section (after reading it):
  • Subgroups (criteria and examples).
  • Problem 2.52(x) and (xi), 2.54(i) (modified), 2.57.
• Write down all questions about the above topics to bring to our (online) lecture. (You can also type them in the file "Questions.tex" in SageMathCloud.) Comments about the videos are welcome!
• Work on the assigned problems for these sections. (See Related Problems below.) You don't need to finish them, but try to work on as many as you can and the bring your questions to class.

 

Related Problems

No problems from these sections will be turned in. (Solutions will be made available immediately after the lecture.) You should try to them all on your own, though, to practice for the Final.

Section 2.3:

2.36 (i) to (v) and (viii) to (ix), 2.37, 2.38, 2.40.

 

Section 2.4:

2.52 (i) to (v) and (x) to (xi), 2.54, 2.55, 2.57.

 

In Class

In class:

• We will discuss the reading and pace.
• I will discuss the main points.
• I will answer questions about the sections covered.
• I will answer any other questions.
• We can work on the HW problems.

 

Outcomes

After the assignment (reading and videos before class) and class, you should:

• know the definition, basic properties and examples of groups;
• know how to prove basic properties of groups (Problems 2.37 and 2.40.);
• know how to translate from multiplicative notation to additive notation;
• know how to compute order of elements (Problem 2.38);
• know how to check if a subset of a group is a subgroup (Problem 2.54);
• know the definition of the alternating group $A_n$;
• know the definition of cyclic groups and how to check if a group is cyclic (Problem 2.52(x) and (xi));
• know Lagrange's Theorem and how to apply it in simple proofs (Problem 2.57).