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October 10th, 2007, 07:10 PM   #1
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order in GL(2,R)

In GL(2,R), let
P=
(0 -1)
(1 1)
Q=
(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.
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October 16th, 2007, 07:47 PM   #2
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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,
(PQ)^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.
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