Understand the problem

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

Source of the problem
TIFR GS 2018 Part A Problem 17
Abstract Algebra
Difficulty Level
Suggested Book
Dummit and Foote

Start with hints

Do you really need a hint? Try it first!

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’s 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?
  • 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.}
  • 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.
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.

Watch the video

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

Partial Differentiation | IIT JAM 2017 | Problem 5

Try this problem from IIT JAM 2017 exam (Problem 5).It deals with calculating the partial derivative of a multi-variable function.

Rolle’s Theorem | IIT JAM 2017 | Problem 10

Try this problem from IIT JAM 2017 exam (Problem 10).You will need the concept of Rolle’s Theorem to solve it. You can use the sequential hints.

Radius of Convergence of a Power series | IIT JAM 2016

Try this problem from IIT JAM 2017 exam (Problem 48) and know how to determine radius of convergence of a power series.We provide sequential Hints.

Eigen Value of a matrix | IIT JAM 2017 | Problem 58

Try this problem from IIT JAM 2017 exam (Problem 58) and know how to evaluate Eigen value of a Matrix. We provide sequential hints.

Limit of a function | IIT JAM 2017 | Problem 8

Try this problem from IIT JAM 2017 exam (Problem 8). It deals with evaluating Limit of a function. We provide sequential hints.

Gradient, Divergence and Curl | IIT JAM 2014 | Problem 5

Try this problem from IIT JAM 2014 exam. It deals with calculating Gradient of a scalar point function, Divergence and curl of a vector point function point function.. We provide sequential hints.

Differential Equation| IIT JAM 2014 | Problem 4

Try this problem from IIT JAM 2014 exam. It requires knowledge of exact differential equation and partial derivative. We provide sequential hints.

Definite Integral as Limit of a sum | ISI QMS | QMA 2019

Try this problem from ISI QMS 2019 exam. It requires knowledge Real Analysis and integral calculus and is based on Definite Integral as Limit of a sum.

Minimal Polynomial of a Matrix | TIFR GS-2018 (Part B)

Try this beautiful problem from TIFR GS 2018 (Part B) based on Minimal Polynomial of a Matrix. This problem requires knowledge linear algebra.

Definite Integral & Expansion of a Determinant |ISI QMS 2019 |QMB Problem 7(a)

Try this beautiful problem from ISI QMS 2019 exam. This problem requires knowledge of determinant and definite integral. Sequential hints are given here.