Re: Covering map of Lie groups
From: Timothy Murphy (tim_at_birdsnest.maths.tcd.ie)
Date: 03/02/05
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Date: Wed, 02 Mar 2005 20:50:56 +0000
Robin Chapman wrote:
>> Does GL(n,C) have normal subgroups of finite order,
>> apart from {I} and {I,-I} ?
>
> Yes, of course, those generated by zI where z is any root
> of unity.
As was pointed out, my remark was slightly inaccurate,
in fact completely wrong!
What I meant to say is that a matrix of finite order d
is necessarily semisimple, ie diagonalisable (over C),
since it satisfies the separable equation x^d - 1 = 0.
It's easy to see from that if the subgroup is normal,
its elements must all be of the form aI.
Incidentally, the homomorphism
T |-> (det T)^e T: GL(n,C) -> GL(n,C)
has quite a nice kernel, for appropriate e.
[Apologies if this has been pointed out.]
-- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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