Non-standard groups of the second type

In [4] it is proved that the multiplicative group Fp⁎ is an automatically non-standard f-subgroup, here called a non-standard f-subgroup of the second type, for suitable f(x)∈Fp[x] whenever p is a prime with p>3 and p≡3(mod4). Here we extend this result to all odd non-Fermat primes p, and give two different proofs. Our first proof is an elaboration of that in [4], while the second proof leads to a general lower bound on the number of f-sequences that represent a non-standard f-subgroup of the second type; in certain very special cases, this lower bound is shown to be exact, although it is known not to be exact in general.