Question

A normal subgroup is ____________

(a) a subgroup under multiplication by the elements of the group

(b) an invariant under closure by the elements of that group

(c) a monoid with same number of elements of the original group

(d) an invariant equipped with conjugation by the elements of original group

This question was addressed to me by my school principal while I was bunking the class.

This intriguing question originated from Groups topic in chapter Groups of Discrete Mathematics

Answer 1

Right option is (d) an invariant equipped with conjugation by the elements of original group

To elaborate: A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group that is, K is normal if and only if gKg-1=K for any g belongs to G Equivalently, a subgroup K of G is normal if and only if gK=Kg for any g belongs to G.Normal subgroups are useful in constructing quotient groups and in analyzing homomorphisms.