Abstract Algebra Answers

3.
a) Let G be a finite group. Show that the number of elements g of G such that g^3 = e is
odd, where e is the identity of G.

1 a b
b) Check if { [ 0 1 c ] | a,b,c belongs to R} is an abelian group w.r.t matrix multiplication.
0 0 1

c) Check whether H={x belongs to R* | x=1or x is irrational} * R and K={x belongs to R* | x>=1} are subgroups of (R*,.).

d) Let U(n)={m belongs to 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 belongs to U(10).

e) Find Z(D2n), where D2n is the dihedral group with 2n elements, [D subscript 2n]
i) when n is an odd integer;
ii) when n is an even integer.

2)
a) Prove that 2^n > 4.n for n>=5.

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

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

d) Check whether the relation R ={(x, y)belongs to N×N| xy is the square of an integer} is
an equivalence relation on N .

show that if E be a finite extension of F then E is
normal extension of F if and only if E is a splitting field of some polynomial over F

If K is a finite extension of F and E is a subfield of K which contains F, then [E:F][K:F].