Answer to Question #203115 in Abstract Algebra for Raghav

Question

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 SUT is an infinite set.

he set {x∈Z| x ≡1(mod30)}is a group with respect to multiplication(mod30).

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 ≅ R and Imf ≅{0}.

Answer 1

Solution:

i) False.

Example: A={1}, B={1,2}

Clearly A ⊆ B, then A × B = {(1,1),(1,2)} $\ne$ B

ii) True.

S is clearly a finite set as they are numbers of rolls in IGNOU.

But T is clearly infinite as there are infinite real numbers between 2.5 and 2.55.

And we know that the union of finite and infinite sets is infinite.

iii) True.

Given set be S$=\{...,-59,-29,1,31,61,91,...\}$

which clearly is a group with respect to multiplication(mod30).

iv) True.

Each element of G/N is a coset aN for some a∈G

Let aN, bN be arbitrary elements of G/N where a,b∈G

Then we have

(aN)(bN)=(ab)N

=(ba)N [Because G is abelian]

=(bN)(aN)

Thus, N is abelian.

v) True.

Example: $f: O\rightarrow O$, where O is zero function, having ker f = R and Im f = {0}