ON A FAMILY OF SUBGROUPS OF THE MULTIPLICATIVE GROUP MOD n

For each pair j , n $j,n$ j comma n with n > j $n>j$ n greater than j , we consider the subgroup of the multiplicative group mod n of residues with order dividing n − j $n-j$ n minus j . We generalise some results in the case j = 1 $j=1$ j equals 1 due to Erdős and the current author [‘On the number of false witnesses for a composite number’, Math. Comp. 46 (1986), 259–279]. The case j = 0 $j=0$ j equals 0 is of particular interest.