Understand the problem

Let G, H be finite groups. Then any subgroup of G × H is equal to A × B for some subgroups A<G and B<H
Source of the problem
TATA INSTITUTE OF FUNDAMENTAL RESEARCH -GS-2018 (Mathematics)  -Part A -Question 9
Topic
Group Theory 
Difficulty Level
Medium 
Suggested Book
Abstract Algebra Dummit and Foote

Start with hints

Do you really need a hint? Try it first!

Consider K \(\leq\) G x H . Now if ( a , b ) \(\to\) K then  \( a^{-1} , b^{-1}\)  \(\to\)  K . Also if ( a , b) , ( p,q )  \(\in\) K then ( ap , bq ) , ( pa , qb )  \(\in\) K  So , G \(|_k\) = { g \(\in\) G | ( g , h ) \(\in\) K  for some h \(\in\) H }  is a subgroup of G . Similarly,  H \(|_k\) \(\leq\) H .
We have arrived to the conclusion that G\(|_k\) \(\leq\) G &  H\(|_k\) \(\leq\) H . Now use this as a fact  to guess the answer . Are you sure that  G\(|_k\)  x  H\(|_k\) = K ?  I mean that who confirms that K can be within as A X B 
Yes this is true that for every g \(\in\) G\(|_K\)  \(\exists\) k s.t  ( g , h )  \(\in\) K . But  G\(|_K\) x  H\(|_K\) contain  ( g , h )  \(\forall h\)    \(\in\)   H\(|_K\) .  This is certainly having a bigger expectation .
So , here is a nice counter example . Take G = \(\mathbb{Z_2}\)  &  H = \(\mathbb{Z_4}\) Consider a cyclic subgroup of G x H ; < ( 1 , 1 ) > = { ( 1 , 1 ) , ( 0, 2 ) , ( 1, 3) , (0,0) } Observe that G\(|_k\) = G &  H\(|_k\) = H But  < ( 1 , 1 ) > \(\neq\)  G x H  Hence , the answer is false.

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

Sequences & Subsequences : IIT 2018 Problem 10

This problem appeared in IIT JAM 2018 whch pricisely reqiures concepts of sequences and subsequences from mathematical field real analysis

Cyclic Groups & Subgroups : IIT 2018 Problem 1

This is an application abstract algebra question that appeared in IIT JAM 2018. The concept required is the cyclic groups , subgroups and proper subgroups.

Acute angles between surfaces: IIT JAM 2018 Qn 6

This is an application analysis question that appeared in IIT JAM 2018. The concept required is the multivarible calculus and vector analysis.

Finding Tangent plane: IIT JAM 2018 problem 5

What are we learning?Gradient is one of the key concepts of vector calculus. We will use this problem from IIT JAM 2018 will use these ideasUnderstand the problemThe tangent plane to the surface $latex z= \sqrt{x^2+3y^2}$ at (1,1,2) is given by \(x-3y+z=0\)...

An excursion in Linear Algebra

Did you know Einstein badly needed linear algebra? We will begin from scratch in this open seminar and master useful tools on the way. The open seminar on linear algebra is coming up on 14th November 2019, (8 PM IST).

Linear Algebra total recall (Open Seminar)

Open Seminar on linear algebra. A review of all major ideas. Even if you have little or no knowledge about Linear Algebra, you may join. Register now.

4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 2

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 1

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.