Question

The set {1, 2, 3, ..., 10, 11} is not a group under multiplication modulo
12 because​

Answer 1

Answer:

Assume that H={1,2,3,...n−1} is a group. Suppose that n is not a prime.

Then n is composite, i.e n=pq for 1<p,q<n−1 . This implies that pq≡0(modn) but 0 is not in H. Contradiction, hence n must be prime.

Conversely, Suppose n is a prime then gcd(a,n)=1 for every a in H. Therefore, ax=1−ny, x,y∈H. So, ax≡1(modn). That is every element of H has an inverse. This conclude that H must be a group since the identity is in H and H is associative