Prove that is isomorphic to
as rings.
(i) The correlation coefficient between ________ ___ _______ ___ ______ ___ ______.
___ ________ _____ ______ ________ ____ ___.
_________ __________ _______ _______ _______ __________ _______ ___ ________ _____.
______ __________.
Get Full Answer on WhatsApp
If G is a group with o(g) < 100 and G has subgroups of order 10 and 25, what is the order of G?
If U (R) denotes the group of units of a ring R, show that U (R1 × R2) = U(R1) × U(R2) for rings R1 and R2.
a) Check whether H = {
x= 1 or x isirrational } and K = { x
x
} are subgroups of
b) Let U() { } n = m ∈ |(m, n) =1,m≤ n . Then U(n) is a group with respect to multiplication modulo n. Find the orders of
for each m ∈ U(10).
c) 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.
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 .
What is the order of
Let F be the ring of all functions from to
w.r.t. pointwise, addition and multiplication. Let S be the set of all differentiable functions in F. Check whether S is
i) a subring of F,
ii) an ideal of F.
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) If A and B are two sets such that A ⊆ B, then A × B = B.
ii) If S is the set of people on the rolls of IGNOU in 2016 and T is the set of real numbers lying between 2.5 and 2.55, then T SU is an infinite set.
iii) The set {x
x=1 1(mod 30)is a group with respect to multiplication (mod 30).
iv) If G is a group with an abelian quotient group G/N, then N is abelian.
v) There is a group homomorphism f with Ker f
and Im f
{0}.
vi) There is a 1 – 1 correspondence between the odd permutations of S35 and the even permutations of S35.
vii) If R is a ring such that , a =− a ∀a ∈ R then R is Boolean.
viii) Given any ring R, there is an ideal I of R such that R/1 is commutative.
ix) If S is an ideal of a ring R and f a ring homomorphism from R to a ring R' , then
x) ‘ring’, as we now define it, was first presented to us by Dedekind.