TIFR2010-Maths-A-2

Question

Which of the following is false?

• A. Any abelian group of order 27 is cyclic
• B. Any abelian group of order 14 is cyclic
• C. Any abelian group of order 21 is cyclic
• D. Any abelian group of order 30 is cyclic

Answer 1

The fundamental theorem of finite abelian groups
Direct products of cyclic groups of coprime order are cyclic.

1. Any abelian group of order 27 is cyclic. (direct products of cyclic groups, of orders 3                                                                     and 9) hence not a product of  co-prime                                                                          
2. Any abelian group of order 14 is cyclic.(direct products of cyclic groups, necessarily of                                                                  orders 2 and 7) hence product of  co-prime 
3. Any abelian group of order 21 is cyclic.(direct products of cyclic groups, necessarily of                                                                  orders 3 and 7) hence product of  co-prime
4. Any abelian group of order 30 is cyclic.(direct products of cyclic groups, necessarily of                                                                  orders 2,3 and 5) hence product of  co-prime

Hence option A