Which one of the following is false? A. The set of all bijective functions on a finite set forms a group under function composition. B. The set {1, 2, ..., p-1} forms a group under multiplication mod p, where p is a prime number. C. The set of all strings over a finite alphabet forms a group under concatenation. D. A subset S != empty of G is a subgroup of the group <G, *> if and only if for any pair of elements a, b in S, a * b^(-1) in S.

GATE 1996 · Discrete Mathematics · Group Theory · medium

Answer: C. The set of all strings over a finite alphabet forms a group under concatenation

1. Validate A, B, D: Bijections on a finite set form a group under composition (A true); nonzero residues mod a prime form a multiplicative group (B true); the one-step subgroup test in D is a correct theorem.
2. Test strings under concatenation: Concatenation is closed, associative and has the empty string as identity, but a non-empty string can never be shortened back to '' by concatenation, so inverses do not exist.