Abstract Algebra Answers

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.

Check whether H={x∈ R^*|x =1or x is irrational}and K={x∈ R^*|x ≥1}are subgroups of (R*,.).

The table below is a Cayley table for the group ({e,a,b,c,d},∗). Fill in the blanks.

e a b c d

_______________________

e e e - - -

a - b - - e

b - c d e -

c - d - a b

d - - - - -

_______________________

Let G be a finite group. Show that the number of elements g of G such that g³= e is

odd, where e is the identity of G.

Give a set of cardinality 5 which is a subset of Z\N.

Give an example, with justification, of a function with domain Z \{2,3}and codomain N.Is this function 1 –1?Is it onto ?Give reasons for your answers.

Which of the following statements are true? Justify your answers. (This means that if you

think a statement is false, give a short proof or an example that shows it is false. If it is

true, give a short proof for saying so.)

(i)There is a 1 – 1 correspondence between the odd permutations of S₃₅ the even

permutations of S_35.

(ii)If R is a ring such that a=−a∀a ∈R, then R is Boolean.

(iii)Given any ring R, there is an ideal I of R such that R/I is commutative.

(iv)If S is an ideal of a ring R and f a ring homomorphism from R to a ring R', then f^-1(f(S))=S.

(v)‘ring’, as we now define it, was first presented to us by Dedekind.

Give the smallest n ∈ N for which An is non-abelian. Justify your answer.