Abstract Algebra Answers

Simplify the following Boolean function:
F = A’C + A’B + AB’C + BC, using K-map?

Which of the following statements are true and which are false? Justify your answer with a short
proof or a counterexample.
i) On the set f1;2;3g, R = f(1;1); (2;2); (3;3)g is an equivalence relation.
ii) No non-abelian group of order n can have an element of order n.
iii) For every composite natural number n, there is a non-abelian group of order n.
iv) Every Sylow p-subgroup of a finite group is normal.
v) If a commutative ring with unity has zero divisors, it also has nilpotent elements.
vi) If R is a ring with identity and u 2 R is a unit in R, 1+u is not a unit in R.
vii) If a and b are elements of a group G such that o(a) = 2, o(b) = 3, then o(ab) = 6.
viii) Every integral domain is an Euclidean domain.
ix) The quotient field of the ring
fa+ibja;b 2 Zg
is C.
x) The field Q(p2) is not the subfield of any field of characteristic p, where p > 1 is a prime.

Check whether the following pairs of elements are associates:
i) 5+4i and 5

Find the gcd of x2+6x+1 and x2+3 in Z7[x].

Find, with justification, all the Sylow subgroups of Z15.

Express f as gq+r in the following cases:
i) f = x4+x+1, g = x2+1 in Q[x].
ii) f = 5x3+2x2+3x+1, g = x+2 in Z7[x].

Calculate (2x+3)(3x+4) in Z5[x]:

a) Check whether S is a subring of R in the following cases:
i) R = Q, S = f a
b 2 Q b is not divisible by 3g.
ii) R is the set of all real valued functions on R and S is the set of linear combinations of the
functions fI;cosnt; sinntg where I : R!R is defined by I(x) = 1 for all x 2 R.

Let G = S4, H = A4 and K = f1; (1 2)(3 4); (1 3)(2 4); (1 4)(2 3)g.
i) Check that H=K = h(1 2 3)Hi
ii) Check that K is normal in H.(Hint: For each h 2 H,h 62 K, check that hK = Kh.)
iii) Check whether (1 2 3 4))H is the inverse of (1 3 4 2)H in the group S4=H.

5) a) Let H = h(1 2)i and K = h(1 2 3)i be subroups of S3. Check that S3 = HK. Is S3 the internal
direct product of H and K? Justify your answer.
b) Let s = 1 2 3 4 5 6 7
2 4 5 6 7 3 1and t = 1 2 3 4 5 6 7
3 2 4 1 6 5 7be elements of S7.
i) Write both s and t as product of disjoint cycles and as a product of transpositions,
ii) Find the signatures of s and t.
iii) Compute ts