Math 360, Fall 2018, Assignment 14

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Read:

1. Section 13.
2. Section 14.

Carefully define the following terms, then give one example and one non-example of each:

1. $G/H$ (also known as the left coset space).
2. Coset multiplication (for cosets of a normal subgroup $H\trianglelefteq G$).
3. Quotient (of a group $G$ by a normal subgroup $H$).
4. Canonical projection (of a group $G$ onto a quotient $G/H$).
5. Homomorphism (often shortened to morphism).
6. Monomorphism.
7. Epimorphism.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning whether coset multiplication is well-defined.
2. Theorem relating monomorphisms to "isomorphic copies" of one group inside another.
3. Theorem constraining where a homomorphism can send the identity element.
4. Theorem constraining where a homomorphism can send the inverse of an element.

Solve the following problems:

1. Section 13, problems 1, 2, 3, 8, 9, and 10.
2. Section 14, problems 1, and 9.
3. Prove that any subgroup of index two is normal. (Hint: there are exactly two left cosets, and one of them--the identity coset--coincides with the subgroup. Also, there are exactly two right cosets, and one of them coincides with the subgroup. Now use one of the five equivalent conditions for normality.)
4. Prove that for any $n$, we have $A_n\trianglelefteq S_n$.
5. In class, we drew the group table for $S_3/A_3.$ Now draw the group table for $S_n/A_n$, for arbitrary $n$.

--------------------End of assignment--------------------