Abstract Algebra Answers

Find two different Sylow 2-subgroups of D12.

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-2 and t2s2.

c) Check whether the following pairs of elements are associates:
i) 5+4i and 5-4i.
ii) 5+4i and -4+5i.
iii) 2x2+4x+6 and x2+2x+3.

Expand the following Boolean functions into their canonical form

f(x,y,z,)=xy+yz'+xz'+x'y ?

f(x,y,z) =xy'+x'y'+xyz ?

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.