1. Let G be a set along with a binary operation ◦ such that
• ◦ is associative
• There exists an element e in G such that e ◦ x = x for all x ∈ G
• For each a ∈ G there exists an element a
0
in G such that a
0 ◦ a = e.
Show that G is a group.
2. Prove that Aut(Q) has a single element.
3. What are all the possible disjoint cycle structures for an element in S_6? How many
distinct elements are there for each cycle structure?
Note: |S6| = 720, so you probably don’t want to write them all down.
4. Prove that the transpositions generate S_n. By this I mean that you can produce all
the elements of S_n using products of transpositions.
5. Show that Z[
√
n] is not a PID for square-free odd n < −2.
Hint 1: You might not want to do this directly…
Hint 2: Once you figure out Hint 1, it might be a good idea to remind yourself how
some of your work in Z[
√
−5] would show this. Can you extend to odd n < −2?
6. An ideal I in a commutative ring R is finitely generated if there exists a_1, . . . a_n ∈ R
such that I = <a_1, . . . , a_n>.
Let R be a commutative ring with unity. Prove that R is a Noetherian ring if and only if every ideal in
R is finitely generated.
7. Let a_1, . . . a_n be elements (not all 0) of an integral domain R. A greatest common
divisor of a_1, . . . a_n is an element d of R such that:
~~~d divides each of the ai
in R,
~~~if c ∈ R and c divides each of the ai
in R then c | d
Let R be a UFD. If a | bc and 1R is a gcd of a and b, prove that a | c.
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