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Show that φ is a group homomorphism

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FINAL OF ALGEBRA

No credit will be given for an answer without reasoning.

1.

(1) [5%] Let F be the additive group of all continuous functions mapping R into R, and let φ : F → R (viewed as an additive group) be given by φ(f ) = R1

0 f (x) dx. Show that φ is a group homomorphism.

(2) [5%] Let µ = (1, 2, 4, 5)(3, 6) in S6. Find the index of the subgroup hµi in S6.

2.

(1) [5%] Find φ(10, 3) for φ : Z × Z → S10 where φ(1, 0) = (1, 7)(6, 10, 8, 9) and φ(0, 1) = (3, 5)(2, 4).

(2) [5%] Let H and K be normal subgroups of a group G. Give an example showing that we may have H ' K while G/H is not isomorphic to G/K.

3.

(1) [5%] Is there a nontrivial (group) homomorphism φ : Z5 → Z? Why or why not?

(2) [5%] If a and b are elements in a group G, show that ab and ba have the same order.

4. Let X be the set of all subgroups of a group G.

(1) [5%] Check that X is a G-set by the map ∗ : G × X → X given by ∗(g, H) = gHg−1. (2) [5%] Let O(H) denote the orbit containing H of the action given in (1). Show that O(H) =

{H} if and only if H is a normal subgroup of G.

5.

(1) [5%] Let M2(Zn) denote the ring of 2 by 2 matrices over the ring Zn. Find the characteristic of M2(Zn).

(2) [5%] Let R be a ring with unity 1. Suppose a ∈ R and a2 = a. Show that 1 − 2a is a unit of R.

6. An element a of a ring R is idempotent if a2 = a.

(1) [5%] Show that the set of all idempotent elements of a commutative ring is closed under multi- plication.

(2) [5%] Find all idempotents in the ring Z × Z6.

7.

(1) [5%] Let D be an integral domain and Q(D) denote the field of quotients of D. Show that Q(D) is isomorphic to D if D is in fact a field.

(2) [5%] Find all zeros of x2+ 1 in the field Z7.

8. Let R[x] denote the ring of polynomials over R. Let N := { f (x) ∈ R[x] | f (2) = 0 }.

(1) [5%] Show that N is an ideal of R[x].

(2) [5%] Is N a maximal ideal of R[x]? Why or why not?

9.

(1) [5%] Show that for p a prime, the polynomial xp+ a in Zp[x] is not irreducible for any a ∈ Zp. (2) [5%] Let R be a ring with unity of prime characteristic p. Show that the map φp: R → R given

by φp(a) = apis a (ring) homomorphism.

10. Show that End(hZ, +i) is isomorphic to the ring hZ, +, ·i.

1

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