Math 360, Fall 2019, Assignment 14

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

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

1. $H\leq G$
2. $H\trianglelefteq G$.
3. $G/H$.
4. $H\backslash G$.
5. Coset multiplication (when $H\triabglelefteq G$).

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

1. Formula for the cardinality of $G/H$.
2. Theorem characterizing when coset multiplication is well-defined.
3. Theorem concerning the properties of coset multiplication ("When $H\trianglelefteq G$, coset multiplication turns $G/H$ into a...").

Solve the following problems:

1. Section 14, problems 1, 9, 24, 30 (hint: use Lagrange's theorem in $G/H$), 31, and 34.

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