Group  ^{List No.}^{Last Name}^{Name}  ^{Student No.}^{Department}  ^{Section}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Instructors}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{BASIC ALGEBRAIC STRUCTURES}^{Midterm II}^{Math 116}^{2008-2009}^{Spring}^{G.E., S.F., M.K.}^{May 14 2009}^{17:40}^{100 minutes}  ^{4 QUESTIONS ON 4 PAGES}^{TOTAL 60 POINTS}^{1}  ^{2}  ^{3}  ^{4}

Show your work ! Partial credits will not be given for correct answers if they are not justified.

\underline{Question 1}

a) Find all subgroups of ℤ_{15}. For each cyclic subgroup find a generator.

b) Find the set ℤ_{15}^{∗} of invertible elements of ℤ_{15} (with respect to multiplication). Why ℤ_{15}^{∗} is a group ? Is it a subgroup of ℤ_{15} ?

c) Which of the elements of ℤ_{15}^{∗} have order 2 ?

d) Prove that group ℤ_{15}^{∗} is not cyclic.

\underline{Question 2}

Consider permutation f = ( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 8 & 7 & 5 & 6 & 4 & 2 & 3 & 1 \end{matrix} ) ∈ S_{9}

a) Find the orbits of f in the set {1, 2, …, 9} and present f as a product of cycles.

b) Find the order of f.

c) Is f an element of the alternating subgroup A_{9} of S_{9} ? Explain why f is not a product of 11 transpositions.

d) Find f^{55}.

\underline{Question 3}

In the following list of examples: ℤ, ℚ, ℤ_{10}, ℤ_{11}, 3ℤ, S_{5}, M_{2}(ℤ), M_{2}(ℝ), ℝ_{∗} = ℝ ∖ {0}, ℤ/3ℤ, ℤ_{8}/2ℤ_{8} a) Which ones are not rings ?

b) Which ones are rings with a unity ?

c) Which ones are non-commutative rings ?

d) Which ones are integral domains ?

e) Which ones are fields ?

\underline{Question 4}

a) Justify that 3ℤ_{6} = {3x | x ∈ ℤ_{6}} is an ideal in ℤ_{6}. Write explicitly the elements of the quotient ring R = ℤ_{6}/3ℤ_{6}.

b) Find the addition and the multiplication tables of R.

c) Is R a ring with a unity (find it, if there is) ? Does it contain zero divisors ?

d) Describe a group isomorphism between ℤ_{6}/3ℤ_{6} and ℤ_{3}. Is it a ring isomorphism ?