My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum

LinkBack Thread Tools Display Modes
October 10th, 2007, 08:10 PM   #1
Joined: Oct 2007

Posts: 7
Thanks: 0

order in GL(2,R)

In GL(2,R), let
(0 -1)
(1 1)
(0 1)
(-1 1).

a)compute to show that P has order 6 and Q has order 4. (and check that the orders cannot be smaller)
b)compute PQ, prove by induction what (PQ)^n is, and show that PQ has infinite order.
bjh5138 is offline  
October 16th, 2007, 08:47 PM   #2
Joined: Mar 2007

Posts: 57
Thanks: 0

I assume by "order" you mean the lowest power you have to raise to get an element to the identity?

For part a)
Compute P^2, P^3, P^4, P^5, P^6 and hopefully you get P^6 = identity in GL_2 (R) while the others are not.
Similarly you can compute Q^2, Q^3, Q^4 and hope you get Q^4 = identity while Q^2 and Q^3 are not.

For part b) compute
PQ =
(1 -1)
(-1 1)

(PQ)^2 = PQPQ =
(2 -2)
(-2 2)

So you should be able to use induction to prove for all n,
(n -n)
(-n n)

Hence (PQ)^n cannot be the identity in GL_2 (R) for any finite n, which means (PQ) does not have finite order.
aptx4869 is offline  

  My Math Forum > College Math Forum > Abstract Algebra

gl2, order

Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Order Adequacy "Q is Order Adequette..." M_Strauss Applied Math 0 October 31st, 2013 10:37 AM
Proof that group of order 2k containts order k subgroup Kappie Abstract Algebra 0 April 22nd, 2012 02:52 PM
Reduction of second order ode to first order Grayham1990 Calculus 2 March 30th, 2012 07:24 AM
Transform 2nd order ODE to two 1st order ODE using matrices Norm850 Calculus 2 March 7th, 2012 05:08 PM
If a has order hk modulo n, then a^h has order k mod n. Jamers328 Number Theory 1 December 2nd, 2007 09:21 PM

Copyright © 2018 My Math Forum. All rights reserved.