Answer to Question #214484 in Abstract Algebra for Priya

Question

If N is normal subgroup of G and a belongs to N, then:
(a) c(a)€N (b) N€c(a)
(c) c(a)={e} (d) None of these

Answer 1

$\text{Since N is a normal subgroup of G and a belongs to N, we know by definition that the}\\\text{set of left cosets is equal to the set of right cosets i.e., ah = ha . Since the centralizer(c(a))}\\\text{of a is the set of elements that commute with a, we have that c(a) is a subset of N. } \\\text{Therefore the answer is a.}$