Math 361, Spring 2014, Assignment 12

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

1. Letter (in a fixed alphabet $S$).
2. Syllable.
3. Word.
4. Concatenation (of two words).
5. Elementary contraction of type 1.
6. Elementary expansion of type 1.
7. Elementary contraction of type 2.
8. Elementary expansion of type 2.
9. Equivalent words.
10. Reduced word.
11. Free group (generated by the alphabet $S$).

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

1. Universal mapping property of the free group (Theorem 39.12)

Solve the following problems:

1. For which alphabet(s) $S$ is $FG[S]$ a finite group?
2. For which alphabet(s) $S$ is $FG[S]$ an abelian group?
3. Section 39, problems 1, 3, and 4.

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