Group Theory Puzzles...

An eccentric group theory professor liked to give puzzles to his students. The puzzles would always be to guess the professor's favourite group (for that year). This particular year, he stated `My favourite group has order eight'. That was the only clue he would give publically, although of course the students would ask him questions after class in an effort to narrow down the choice.

One particularly bright student was slightly delayed on his way to the professor's office, and as a result, there were two students there ahead of him wanting to ask about the group. As they chatted, the students made the following remarks:

 ``I'm going to ask him how many elements of order two the group has!'' the first one said

 ``Well, I'm going to ask him about its elements of order four!'' said the other

 The third student watched, as the first one went in, and then a short time later came out, with a long face. ``Well, he told me, but it didn't help'' he confessed. The story was repeated - the second student went in to the professor's room and then came out with a sad look. ``Well, looks like you get a chance after all'', they said to the third.

 ``A chance?'' he said ``Why there's no chance involved! I know the group!'' and with these words, he strolled in to collect his prize.

 What is the answer? 

Two students were discussing one day:
  • Is that professor still giving prizes to people who guess his favourite group?
  • Yes - this year it was order twelve
  • Really? Were there any takers for the prize?
  • Just one, I think. He asked the professor two questions before he won.
  • What were the questions?
  • I can't recall exactly - I think the first one was `Does the group have any elements with such and such an order?'
  • `Such and such'?
  • I can't remember what order he was asking about. Anyway, the professor's answer didn't help him much. He had to ask a different question.
  • What was his second question?
  • It was almost the same - `Does the group have any elements with order so and so?'.
  • What happened then?
  • Well, that clinched it. When the professor answered, he knew immediately what the group was, and so he won the prize.
  • Good on him. Was he happy with the prize?
  • A bit, I guess - but mostly he was complaining he hadn't asked the questions the other way around. He wanted to get the prize using only one question, you see...
What is the answer?