Abstract Algebra

Abstract Algebra

1. Which of the examples 1-20 are familiar to you?

Example 1, 2, 4, 6, 8, and 13.

2. What do you think of grouping all these disparate kinds of mathematical objects under one name?

So far it makes sense to me. Attempting to group things and study the abstract consequences of these groupings. I could see myself having issues "viewing the groups as abstract entities rather than argue by example," (Gallian, 48).

3. Why is the property that every element has an inverse important?

In abstract algebra the inverse works to 'undo' the effect of a combination with another given element. IN a way, it generalizes the concept of negation.

4. Can you think of a binary operation where "cancellation" (Thm 2.2) holds, but where not every element has an inverse?

Not initially, no! But after researching it, the "Cancellative semigroup." This group has the cancellation property, but it doesn't require its elements to have an inverse.

5. Explicit questions or comments I found interesting or confusing

a. I researched the answer to question 4, but still wasn't able to come up with a true binary operation where those conditions exist.

Homework 3A

1. Is D4 Abelian or not? Explain.

Abelian is another word for commutative, or that ab=ba for all choices of group elements a and b. D4 is non-Abelian because for instance, the operation D'H=R90 but HD'=R270. Therefore, D'H≠HD' and therefore D4 is non-Abelian.

2. How does Gallian's use of the word "symmetry" compare with whatever take on the word you had before?

I previously would think of symmetry as two things being the same on either side of a line.

Gallian uses the word symmetry similar to the Greek word symmetros, or "of like measure." He states many different types of symmetry. Plane symmetry being a function from the plane to itself that carries F onto F and preserves distances. The symmetry group of a plane figure is the set of all symmetries of the figure. He then states, like my take on the word, a rotation about a line in three dimensions is symmetry. Lastly, any translation of a plane or 3D space is also symmetry.

3. Why does Clark use the word "binary" in the term "binary operation"?