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Multiplicative group from fields: TIFR GS 2018 Part A Problem 17

Understand the problem

The multiplicative group \(F^*_7\) is isomorphic to a subgroup of the multiplicative group \(F^*_{31}\). 

Start with hints

Hint 1
We will write them as (Z/7Z)* and (Z/31Z)* respectively instead of the notations used.
  • Observe that (Z/7Z)* has order 6 and (Z/31Z)* has order 30.So there is a possibility that (Z/7Z)* is a subgroup of (Z/31Z)* by Lagrange‚Äôs Theorem.
  • So we need to go into the structure of the groups to solve this problem.Hence we proceed!
  • Let us investigate the group (Z/7Z)*.It consists of {1,2,3,4,5,6 mod 7}.Observe that 3 mod 7 generates the group.
  • So naturally the next question is that whether (Z/31Z)* rather is there any general result?
Hint 2
  • In fact the following theorem is true and describes the cyclicity (Z/nZ)* to some extent.
  • Theorem: If p is a prime then (Z/pZ)* is cyclic. (Check!) {Check the bonus question for the complete characterization of cyclicity of (Z/nZ)* done by Gauss.}
Hint 3
  • So (Z/7Z)* and (Z/31Z)* are cyclic groups of order 6 and 30 respectively with generators say A and B respectively.
  • Now take the element \(B^5\).The following Lemma describes its order.
  • Lemma: If g is the generator of the cyclic group of order n. Then \(g^k\_ has order n/gcd(n,k).(Check !)
  • So \(B^5\) has order 6 and hence it is isomorphic to (Z/7Z)*.
  • Hence the answer is True.
Hint 4
Bonus Problem:
  • Theorem: The group (Z/nZ)* is cyclic if and only if n is $1, 2, 4,p^k$ or 2.$p^k$, where p is an odd prime and k > 0. This was first proved by Gauss. (Wow!)
Solve and Salvage if Possible.

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