Algebra I Final Project

Name:

Instruction: In the following questions, you should work out the solutions independently and write them down in a clear and concise manner. Staple this page as the cover sheet of your project.

1. (30 pts) (a) A pointed set is a pair (S, x) with S a set and x ∈ S. A morphism of pointed sets (S, x) → (S0, x0) is a triple (f, x, x0), where f : S → S0 is a function such that f(x) = x0. Show that pointed sets form a category.

(b) Show that in the category S∗ of pointed sets products always exist; describe them.

(c) Show that in S∗ every family of objects has a coproduct (often called a “wedge product”); describe this coproduct. 2. (20 pts) Let F be the free group on a set X and G the free group on a set Y . Let F 0 be the subgroup of F generated by {aba−1b−1 | a, b ∈ F } and similarly for G0. Prove the following claims:

0 0 (a) F C F , and F/F is a free abelian group of rank |X|. (b) F ' G if and only if |X| = |Y |. In particular, if F is also a free group on a set Z, then |X| = |Z|. 3. (20 pts) Every subgroup and every quotient group of a nilpotent group is nilpotent.

4. (10 pts) Let H be a subgroup of a finite group G. Suppose that P is a Sylow p-subgroup of H. If NG(P ) ⊆ H, show that P is a Sylow p-subgroup of G.

5. (40 pts) Let R be a principal ideal domain.

(a) Every proper ideal is a product P1P2 ··· Pn of maximal ideals, which are uniquely determined up to order. (b) An ideal P in R is said to be primary if ab ∈ P and a 6∈ P imply bn ∈ P for some n. Show that P is primary if and only if for some n, P = (pn), where p ∈ R is prime (= irreducible) or p = 0.

ni (c) If P1,P2, ··· ,Pn are primary ideals such that Pi = (pi ) and the pi are distinct primes, then P1P2 ··· Pn = P1 ∩ P2 ∩ · · · ∩ Pn. (d) Every proper ideal in R can be expressed (uniquely up to order) as the intersection of a finite number of primary ideals. 6. (10 pts) If f : A → A is an R-module homomorphism such that ff = f, then

A = Ker f ⊕ Imf.

7. (20 pts) Let

A1 / A2 / A3 / A4 / A5

α1 α2 α3 α4 α5      B1 / B2 / B3 / B4 / B5 be a commutative diagram of R-modules and R-module homomorphisms, with exact rows. Prove that:

(a) α1 an epimorphism and α2, α4 monomorphisms ⇒ α3 is a monomorphism;

(b) α5 a monomorphism and α2, α4 epimorphisms ⇒ α3 is an epimorphism.

1