Question

Which one of the following is false ?

• A. A subset $s\ne \varphi$ of G is a subgroup of the group $<G,\ast >$ if and only if for any pair of elements
  $a,bϵs,a\ast b^{-1}ϵs$
• 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 biijective functions on a finite set forms a group under function composition.
• D. The set of all strings over a finite alphabet $\sum$ forms a group under concatenation.

Answer 1

The correct option is D The set of all strings over a finite alphabet $\sum$ forms a group under concatenation.

In option (c), the set of all strings over a finite alphabet $\sum$ doesn't forms a group under concatenation because the inverse of a string doesn't exist with respect to concatenation.