Question about discarding the second register in the standard approach of hidden subgroup algorithm
Discarding in this context can be carried out by performing a Partial Trace. The system you're interested in is that of the first register. If you want to know only its state, as opposed to the combined state of the two registers, then you take the density matrix of the whole system and trace out the second register.  The resultant state will, in this case, be mixed.  What you're seeing at work is more or less purification in reverse. The resultant mixed state is the reduced state of some other pure system, namely the system comprised of the two registers.