Abstract Algebra Answers

a) Use the Fundamental Theorem of Homomorphism to prove that Z/12~ (g) iff g is an
element of order 12 in a group (G; ). (7)
b) Obtain two distinct elements of Z/7Z, and two distinct subgroups of Z/7Z. (

Let S = { matrix [ a 0] | a,b ∈ Z } .
[0 b ]

i) Check that S is a subring of M(subscript2)(R) and it is a commutative ring with identity.
ii) Is S an ideal of M(subscript2)(R)? Justify your answer.
iii) Is S an integral domain? Justify your answer.
iv) Find all the units of the ring S.
v) Check whether
I = { matrix [ a 0] | a,b ∈ Z, 2 | a } .
[0 b]
is an ideal of S.
vi) Show that S is congruent to Z×Z where the addition and multiplication operations are componentwise addition and multiplication.

Let σ = (a1 a2 ...ak) ∈ Sn be a cycle let τ ∈ Sn.
i) Check that τ σ τ^−1 = (b1 b2 ···bk), where τ (ai) = bi.
ii) Use the above result to compute τ σ τ^−1 where σ and τ are as in part b).

Show that the map f : Z+iZ → Z2, defined by f(a+ib) = (a−b) (mod 2), is an onto ring homomorphism. Describe kef f. Is it a maximal ideal? Justify your answer.

Factorise 10 in two ways in Z[under-rootof -6]. Hence, show that Z[under-rootof -6] is not a UFD.

Factorise 10 in two ways in Z[p

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 ?