Abstract Algebra Answers

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

Find Z(D2n), where D2n is the dihedral group with 2n elements,

i) when n is an odd integer;

ii) when n is an even integer.

Let U(n)={m∈N|(m, n) =1,m≤ n}. Then U(n) is a group with respect to

multiplication modulo n. Find the orders of <m> for each m∈U(10).

Check if matrix

1 a b

[A ]= 0 1 c

0 0 1

Where a,b,c∈R

is an abelian group with respect to matrix multiplication.