Understand the problem

The permutation group \(S_{10}\) has an element of order 30.
Source of the problem
TIFR 2018 Part A Problem 23
Topic
Group Theory
Difficulty Level
Easy
Suggested Book
Dummit and Foote

Start with hints

Do you really need a hint? Try it first!

Consider S={1,2,…,10}.\(S_{10}\) be the permutation group on S.
What will you do if one asked for a subgroup of order 3!=6?
  • We all would have taken the subgroup of \(S_3\) embedded in \(S_{10}\) right? Call the subgroup H taking identity transformation on {4,5,6,..10} and embedding in {1,2,3}.
What do you do if one asked for a subgroup of order 5?
  • We will try to form a cyclic subgroup of order 5. We need to find a generator. Can you see it?
  • Keeping {1,2,.,5} same and taking 5 elements {6,7,..,10} and observe the permutation \(i\mapsto i+1\) for i=6,7,8,9 and \(10\mapsto 7\).Take the subgroup generated by this element.Observe this is a cyclic subgroup of order 5.Call this subgroup K.
  • Now any idea how to combine this?
  • Observe that HK is a set of 30 elements. Does it seem HK is a subgroup?
  • Lemma: H and K are two subgroups of G. HK is a subgroup of G iff KH=HK.
  • Using the lemma prove that HK is really a subgroup of \(S_{10}\) of order 30.
  • Observe that the selection of disjoint elements of H and K is the main reason behind this!
  • Hence the answer is True.

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.