Which one is true?a)Every quotient group of a group is abelian and its converse is also true.b)Every quotient group of a group is abelian but its converse is not truec)Every quotient group of an abelian group is abelian and its converse is also true.d)Every quotient group of an abelian group is abelian but the converse is not true.Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation:

In order to understand the answer, we need to first define what a quotient group is and what it means for a group to be abelian.

Quotient Group:
Given a group G and a normal subgroup N of G, the quotient group G/N consists of the cosets of N in G with the group operation defined as (aN)(bN) = (ab)N, where aN and bN are cosets of N in G. In other words, the elements of G/N are the subsets of G that are formed by taking each element of G and multiplying it by the elements of N.

Abelian Group:
A group is said to be abelian (or commutative) if the group operation is commutative. In other words, for all elements a and b in the group, ab = ba.

Now, let's analyze each option:

a) Every quotient group of a group is abelian and its converse is also true:
This statement is not true. The quotient group of a group is not necessarily abelian. It may or may not be abelian, depending on the group and the normal subgroup chosen.

b) Every quotient group of a group is abelian but its converse is not true:
This statement is also not true. The quotient group of a group is not always abelian. There exist examples where the quotient group is not abelian.

c) Every quotient group of an abelian group is abelian and its converse is also true:
This statement is not true. While it is true that every quotient group of an abelian group is abelian, the converse is not true. There exist examples of non-abelian groups whose quotient groups are abelian.

d) Every quotient group of an abelian group is abelian but the converse is not true:
This statement is true. If G is an abelian group and N is a normal subgroup of G, then the quotient group G/N is also abelian. However, the converse is not true. There exist non-abelian groups whose quotient groups are abelian.

Conclusion:
Option D is the correct answer. Every quotient group of an abelian group is abelian, but the converse is not true.

Similar Mathematics Doubts

(i) Every Quotient group of an abelian group is abelian(ii) Every Quotient group of a cyclic group is not cyclic.

Consider the statements:S1: Let G be an abelian group of order n if for every divisior m of n there exist a subgroup of G of order m, thenG is cyclic.S2,: Let G be a group. If every proper sub group of G is cyclic then G is abelian.Which of the following is true.

Which one of the following is false

Which statements is/are true ?

Choose the correct option

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