Group Theory Clarification

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z4lis
Posts: 767
Joined: Mon Mar 03, 2008 10:59 pm UTC

Group Theory Clarification

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?
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hemb
Posts: 27
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Re: Group Theory Clarification

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.

Qoppa
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Re: Group Theory Clarification

[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
Posts: 155
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Location: Bristol, UK

Re: Group Theory Clarification

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

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.