Select Page

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
Medium
Suggested Book
Dummit and Foote

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.

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

Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 2a

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

Data, Determinant and Simplex

This problem is a beautiful problem connecting linear algebra, geometry and data. Go ahead and dwelve into the glorious connection.

Problem on Integral Inequality | ISI – MSQMS – B, 2015

Try this problem from ISI MSQMS 2015 which involves the concept of Integral Inequality and real analysis. You can use the sequential hints provided to solve the problem.

Inequality Problem From ISI – MSQMS – B, 2017 | Problem 3a

Try this problem from ISI MSQMS 2017 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

Problem on Natural Numbers | TIFR B 2010 | Problem 4

Try this problem of TIFR GS-2010 using your concepts of number theory and congruence based on natural numbers. You may use the sequential hints provided.

Definite Integral Problem | ISI 2018 | MSQMS- A | Problem 22

Try this problem from ISI-MSQMS 2018 which involves the concept of Real numbers, sequence and series and Definite integral. You can use the sequential hints

Inequality Problem | ISI – MSQMS 2018 | Part B | Problem 4

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality and Combinatorics. You can use the sequential hints provided.

Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 4b

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

Positive Integers Problem | TIFR B 201O | Problem 12

Try this problem of TIFR GS-2010 using your concepts of number theory based on Positive Integers. You may use the sequential hints provided.

CYCLIC GROUP Problem | TIFR 201O | PART A | PROBLEM 1

Try this problem from TIFR GS-2010 which involves the concept of cyclic group. You can use the sequential hints provided to solve the problem.