Can there be a homomorphism from ? Give reasons for your answer
To determine whether there exists a homomorphism from \(\mathbb{Z}_8 \oplus \mathbb{Z}_2\) onto \(\mathbb{Z}_4 \oplus \mathbb{Z}_4\), we analyze their order and structure. ### Step 1: Compute the Orders The order of a group is the number of elements in it. - \(\mathbb{Z}_8 \oplus \mathbb{Z}_2\) has: \[ |\mathbb{Z}_8 \oplus \mathbb{Z}_2| = 8 \times 2 = 16 \] - \(\mathbb{Z}_4 \oplus \mathbb{Z}_4\) has: \[ |\mathbb{Z}_4 ___ ______ ____ ________ ____ ___ ________ _________ ________ ____ ______ ________.
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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.
Prove that is isomorphic to
as rings.
a) Prove that n 2 4 n > for n ≥ 5.
b) Give an example, with justification, of a function with domain \{ 2,3 }and codomain
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
d) Check whether the relation R ={(x, y)∈ ×
|xyis the square of an integer} is an equivalence relation on
.
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.
Let R be a ring, I an ideal of R, J an ideal of I. Show that if J has a unity, then J is an ideal of R. Also give an example to show that if J does not satisfy this condition it need not be an ideal of R.
Prove that every ideal I of a ring R is the kernel of a ring homomorphism of R.
If G is a group with o(g) < 100 and G has subgroups of order 10 and 25, what is the order of G?