Math 360, Fall 2020, Assignment 14

I must study politics and war, that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

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

1. $H\leq G$
2. $H\trianglelefteq G$.
3. Coset multiplication (when $H\trianglelefteq G$).
4. Canonical projection (from $G$ onto $G/H$).

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

1. Formula for the cardinality of $G/H$ (we did not state this in class, but think carefully about the picture that accompanied the proof of Lagrange's Theorem).
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: this is similar to the proof last week that $a^{\left\lvert G\right\rvert}=e$; this time use Lagrange's theorem in $G/H$), 31, and 34.

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