Group Theory Clarification

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z4lis
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Group Theory Clarification

Postby z4lis » Tue Sep 08, 2009 2:40 am UTC

A problem I'm working with gives a group, a normal subgroup, and then asks me to identify G\H by using the first isomorphism theorem. What, exactly, does it mean by "identify"? Just fine another, better-known group (right now I've got the reals under multiplication) that G\H is isomorphic to? Should I also mention what the cosets of H look like?
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

hemb
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Re: Group Theory Clarification

Postby hemb » Tue Sep 08, 2009 3:08 am UTC

I would probably do both, just to be safe. Say what G\H is isomorphic to, as well as a description of what the group "looks" like.

Of course, the easier thing to do is just e-mail the professor and ask what kind of answer he's looking for.

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Qoppa
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Re: Group Theory Clarification

Postby Qoppa » Tue Sep 08, 2009 7:00 am UTC

[imath]G/H[/imath], not [imath]G\backslash H[/imath]. [imath]G\backslash H[/imath] would be the group resulting from their set difference or something (if it's even a group).

Using the first isomorphism theorem suggests to me finding a homomorphism [imath]\phi[/imath] such that [imath]\ker \phi = H[/imath]. Then [imath]G/H[/imath] could be 'identified' as the image of [imath]\phi[/imath].

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majikthise
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Re: Group Theory Clarification

Postby majikthise » Tue Sep 08, 2009 5:38 pm UTC

Qoppa wrote:[imath]G/H[/imath], not [imath]G\backslash H[/imath]. [imath]G\backslash H[/imath] would be the group resulting from their set difference or something (if it's even a group).

Using the first isomorphism theorem suggests to me finding a homomorphism [imath]\phi[/imath] such that [imath]\ker \phi = H[/imath]. Then [imath]G/H[/imath] could be 'identified' as the image of [imath]\phi[/imath].

Which it clearly wouldn't be, due to the lack of an identity element.


Erm, I have nothing real to add. Sorry.
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t0rajir0u
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Re: Group Theory Clarification

Postby t0rajir0u » Tue Sep 08, 2009 6:31 pm UTC

It would probably help if you just told us what the problem is. We shouldn't solve it for you, but it would be easier to see what the expected form of the answer is.


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