Math 360, Fall 2013, Assignment 7

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

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

1. Left congruence (i.e. the relation $\sim_L$ discussed on page 97 of the text).
2. Right congruence.
3. Left coset.
4. Right coset.
5. Index of a subgroup.

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

1. Lagrange's Theorem.
2. Theorem concerning the indices of nested subgroups (Theorem 10.14 in the text).

Solve the following problems:

1. Section 10, problems 1, 7, 13, 15, 22, and 24.

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