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 Post subject: Separating groups into isomorphism classes.Posted: Sat, 3 Mar 2012 06:51:30 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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for #2. i know what all of them look like except for Z3 X Z2. How would i generate a group table for that one??

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sat, 3 Mar 2012 08:30:19 UTC
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DgrayMan wrote:

for #2. i know what all of them look like except for Z3 X Z2. How would i generate a group table for that one??

It's a product, you just list the pairs and add them in coordinates. I'm a bit confused though, none of those questions, in particular not #2, reference Z2 OR Z3...

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sat, 3 Mar 2012 19:52:26 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
Oops wrong pic!

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sat, 3 Mar 2012 22:02:18 UTC
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DgrayMan wrote:
Oops wrong pic!

As Shadow said, nothing particularly difficult about ...it is the direct product of two prime order (cyclic) groups...that tells you a lot right there. For instance, the order of each element is the least common multiple of the orders of each "component" of the element. That implies, for instance, that this group has order 6.

The Cayley table is then easy to construct.

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sat, 3 Mar 2012 22:40:43 UTC
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Justin wrote:
DgrayMan wrote:
Oops wrong pic!

As Shadow said, nothing particularly difficult about ...it is the direct product of two prime order (cyclic) groups...that tells you a lot right there. For instance, the order of each element is the least common multiple of the orders of each "component" of the element. That implies, for instance, that this group has order 6.

The Cayley table is then easy to construct.

Since it's listed as a product, he doesn't need to worry about whether or not the original orders, and he can get order six just by multiplying the orders of the original group, it doesn't have to be cyclic for that to work.

To the op: You know the elements are (0,0), (0,1), (1,0), (1,1), (2,0), and (2,1), the table is easy, just add in components.

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 00:33:12 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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so this is still using addition modulo 3 = {0 ,1 ,2} ? so (0,1)+(0,1)=(0,2) Or does it mean only 0 is mod3 and 1 is mod2? in (0,1), which in that case would make it (0,1)+(0,1)=(0,0)

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 00:41:25 UTC
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DgrayMan wrote:
so this is still using addition modulo 3 = {0 ,1 ,2} ? so (0,1)+(0,1)=(0,2) Or does it mean only 0 is mod3 and 1 is mod2? in (0,1), which in that case would make it (0,1)+(0,1)=(0,0)

The first component adds modulo 3 and the second adds modulo 2.

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 00:43:15 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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Got it! phew!!!

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 00:53:07 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
By inspection, none of them look isomorphic? is this right??

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 01:22:47 UTC
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DgrayMan wrote:
By inspection, none of them look isomorphic? is this right??

No, that's not quite right. Now is the time to use Justin's advice and note that the element (1.1) in Z3 x Z2 maps to the element 1 in Z6 is an isomorphism.

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 01:24:34 UTC
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Since it's listed as a product, he doesn't need to worry about whether or not the original orders, and he can get order six just by multiplying the orders of the original group, it doesn't have to be cyclic for that to work.

True, but I think you can only multiply the orders when they are relatively prime...correct?

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 01:28:33 UTC
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Justin wrote:
Since it's listed as a product, he doesn't need to worry about whether or not the original orders, and he can get order six just by multiplying the orders of the original group, it doesn't have to be cyclic for that to work.

True, but I think you can only multiply the orders when they are relatively prime...correct?

Nope, as a set is just the cartesian product of and , so the orders multiply no matter what they are. You are probably thinking of ways to combine them and generators when they're abelian to find new generators.

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 01:41:55 UTC
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Justin wrote:
Since it's listed as a product, he doesn't need to worry about whether or not the original orders, and he can get order six just by multiplying the orders of the original group, it doesn't have to be cyclic for that to work.

True, but I think you can only multiply the orders when they are relatively prime...correct?

Nope, as a set is just the cartesian product of and , so the orders multiply no matter what they are. You are probably thinking of ways to combine them and generators when they're abelian to find new generators.

Ah, right...I guess I'm thinking of something related to the Fundamental Theorem of Abelian Groups(?)
The one where you characterize the (finite) Abelian groups as direct sums up to isomorphism...

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 01:44:21 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Location: Austin, TX
Justin wrote:
Justin wrote:
Since it's listed as a product, he doesn't need to worry about whether or not the original orders, and he can get order six just by multiplying the orders of the original group, it doesn't have to be cyclic for that to work.

True, but I think you can only multiply the orders when they are relatively prime...correct?

Nope, as a set is just the cartesian product of and , so the orders multiply no matter what they are. You are probably thinking of ways to combine them and generators when they're abelian to find new generators.

Ah, right...I guess I'm thinking of something related to the Fundamental Theorem of Abelian Groups(?)
The one where you characterize the (finite) Abelian groups as direct sums up to isomorphism...

Yes you are, and I suspected as much. Don't worry, this is a common mistake when thinking about these things before you really get them all the way down.

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 Post subject: Re: Separating groups into isomorphism classes.Posted: Sun, 4 Mar 2012 01:51:15 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 21 May 2003 04:27:18 UTC
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Justin wrote:
Ah, right...I guess I'm thinking of something related to the Fundamental Theorem of Abelian Groups(?)
The one where you characterize the (finite) Abelian groups as direct sums up to isomorphism...

Yes you are, and I suspected as much. Don't worry, this is a common mistake when thinking about these things before you really get them all the way down.

LOL...I've been working on getting that for the past six years...some things in math are just harder to digest than others.

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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