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 Post subject: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 01:35:31 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182

For both of these Ive figured of the homomorphism with the f(ab)=f(a)f(b).
But in describing the kernel, I'm a little confused. i know kernel is the set k={x in G: f(x)=e_H}, for f:G-->H

So for 1.)i was thinking k={f in F(R):, o(f)=0, or f(0)=0}
For 2.) all i know is the identity is the identity matrix.

So what would the kernels be?

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 01:42:58 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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DgrayMan wrote:

For both of these Ive figured of the homomorphism with the f(ab)=f(a)f(b).
But in describing the kernel, I'm a little confused. i know kernel is the set k={x in G: f(x)=e_H}, for f:G-->H

So for 1.)i was thinking k={f in F(R):, o(f)=0, or f(0)=0}
For 2.) all i know is the identity is the identity matrix.

So what would the kernels be?

You've already describe the kernel for 1, I don't see what the problem there is.

For 2, the identity is not the [2x2] identity matrix, it's the number 1, and iyou were talking about then that's irrelevant. You just need matrices so that ad-bc=1, which is another name for

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 01:47:16 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
what is SL_2(R)? ive never seen that before

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 01:49:10 UTC
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DgrayMan wrote:
what is SL_2(R)? ive never seen that before

It's the kernel of that map. Like, no circular reasoning, it's defined that way.

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 02:35:02 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
i always feel dumb asking but i'm gonna,, but is that the same as 1/(ad-bc)? OR are you saying there are matrices whose determinants are always 1, and that is the kernel?

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 02:35:45 UTC
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DgrayMan wrote:
i always feel dumb asking but i'm gonna,, but is that the same as 1/(ad-bc)? OR are you saying there are matrices whose determinants are always 1, and that is the kernel?

By definition the kernel is things that map to 1...

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 02:51:04 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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so i can describe the kernel k as k={f in G, f(A)=ad-bc=1}?

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 02:56:37 UTC
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DgrayMan wrote:
so i can describe the kernel k as k={f in G, f(A)=ad-bc=1}?

huh? don't you mean ?

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 03:04:53 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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aren't i describing the function f though?

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 03:07:21 UTC
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DgrayMan wrote:
aren't i describing the function f though?

You're describing its kernel, but that's a subset of , and certainly "the set of f in G..." is gibberish when G doesn't have f's and there is only one f in the first place.

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 Post subject: Re: Examples of Homorphisms of Infinite GroupsPosted: Thu, 5 Apr 2012 03:10:04 UTC
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ok makes sense.

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