Question

If F is a free semigroup on a set S, then the concatenation of two even words is ________

(a) a semigroup of F

(b) a subgroup of F

(c) monoid of F

(d) cyclic group of F

This question was posed to me at a job interview.

This is a very interesting question from Group Axioms topic in chapter Groups of Discrete Mathematics

Answer 1

Right choice is (b) a subgroup of F

The explanation is: Let F be the free semigroup on the set S = {m,n}. Let, E consist of all even words, i.e, words with even length and the concatenation of two such words is also even. Thus E is a subsemigroup of F.