Find all finitely generated groups that have no characterist

Find all finitely generated groups that have no characteristic subgroups other than A and O where As the entire group and O is the identify (Trivial group).

Solution

If order of the group \'A\' = p , prime number

then the group (A, * ) will have only improper subgroups {0} , A

Example the groups (Z_n  +_n ) is agroup if nis a prime number

n= 2,3 ,5,7,11,13,17,19 ----------

then the group (Z_{n ,}+_n) will not have proper subgroups