Question

For which integers n such that 3<= n <=11 is there only one group of order n (up to isomorphism)?

(A) For no such integers n

(B) For 3, 5, 7, and 11 only

(C) For 3, 5, 7,9, and 11 only

(D) For 4, 6, 8, and 10 only

(E) For all such integers n

Answer 1

The correct option is (B)

Because if n is a prime number P, then any group is isomorphic to Zp. Since 3,5,7,11 are prime numbers then upto isomorphism they are isomorphic to only Z3, Z5, Z7, Z11.

But there are 2 groups of order 4 upto isomorphism, namely Z4, K4. (klien's 4 group).

There are 2 groups of order 6, namely S3 and Z6.

There are 3 groups of order 8, namely, Z8, D4, Z4xZ2.

There are 2 groups of order 9, namely, Z9, Z3xZ3.

SO option B is correct option (justied)