Group  ^{List No.}^{Last Name}^{Name}^{Student No.}^{Department} ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{Math 116 - Basic Algebraic Structures}^{Midterm II}^{Math 116}^{2005 - 06}^{AB, SF, T^{}Ö}^{2.5.2006}^{17:40}^{80 minutes}  ^{4 QUESTIONS ON 4 PAGES}^{TOTAL 30 POINTS}^{1}  ^{2}  ^{3}  ^{4}

Question 1. (2+2+1+2=7 points) (a) List all 4-cycles, 3-cycles and 2-cycles in S_{4}. (Write them in cycle notation.)

(b) Write other elements of S_{4} as a product of disjoint cyles (in cycle notation).

(c) Express (1, 2, 3, 4) as a product of 2-cycles.

(d) List all elements of A_{4} as products of 2-cycles.

Question 2. (1+2+1+2+2+1=9 points) Set (G, ·) = (Z_{9}, +) and H = ⟨[3]⟩.

(a) List all cosets of H in G.

(b) Write all elements of each coset.

(c) Explain why H is normal in G.

(d) Write the elements of G/H explicitly.

(e) Make a multiplication table for G/H.

(f) Show that G/H is isomorphic to H.

Question 3. (2+1+1+2+2=8 points) Let S = \{[ \begin{matrix}a & b \\ 0 & c \end{matrix} ] : a, b, c ∈ R\}.

(a) Show that S is a subring of M_{2}(R).

Justify your answers below. Only a yes or no answer will get zero credits.

(b) Does S have a unity? If so, find it.

(c) Is S commutative?

(d) Does S have any zero divisors?

(e) Is S a field?

Question 4. (3+3=6 points) (a) Let R be a commutative ring with unity 1 and Θ : R → R a ring isomorphism. Prove that Θ(1) = 1. (Warning: Note that elements of R may not have multiplicative inverses.)

(b) If Θ : Q → Q is a ring isomorphism on rational numbers, then show that Θ is the identity map.