Abstract Algebra Answers

Define f:Z→Z_m × Z_n :f (x)=(xmodm,xmod n),m,n∈N.

i) If (m, n) = (3, 4), find Ker f.

ii) If (m, n) = (6, 4), find Ker f.

iii) What can you generalize about Ker f from (i) and (ii)?

Can there be a homomorphism from Z₈ ⊕Z₂ onto Z₄⊕Z₄ ? Give reasons for your

answer.

Prove, by contradiction, that A₄ has no subgroup of order 6.

Check whether the subgroup of reflections and subgroup of rotations in D_2n is normal in D_2n or not. (Note that D_2n is the group of symmetries of an n-gon.)

what is the order of

i) 14 in Z₂₄/ _

<8>?

ii) (Z₁₀⊕U(10)/<2,9>?

Show that in a group G of odd order, the equation x²= e has a unique solution.

Further, show that x2= g has a unique solution ∀g∈G,g ≠e .

If G is a group with o(g) < 100 and G has subgroups of order 10 and 25, what is the

order of G?

Obtain the left cosets of V₄= {e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}in A₄.

Prove that Z[√2] is isomorphic to

Matrix H = a 2b

b a

Where a,b∈Z as rings.

Find Z(D_2n), where D_2n is the dihedral group with 2n elements,

i) when n is an odd integer;

ii) when n is an even integer