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In the problems involving two algebraic systems, for eg.,$\langle S,*\rangle$ and $\langle P,\bigoplus\rangle$ where the sets $S=\{a,b,c\}$ and $P=\{1,2,3\}$. Here we have to check whether they both are isomorphic or not. While solving, they take values as $g(a)=3$, $g(b)=1$ and $g(c)=2$ and prove the systems as isomorphic. If I try other combination of values, it doesn't satisfy isomorphism. Then, on what basis these values are chosen(a=3,b=1,c=2)?

Kindly check out this Pg. 234 for the definitions of the operations.

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We'll need more information than this - how are the operations $*$ and $\oplus$ defined? – Matthew Pressland Aug 23 '12 at 15:20
@MattPressland I have edited the question. Please check it out. – Gomathi Aug 23 '12 at 15:53
I've removed algebra tag, since we don't use algebra tag anymore, see meta for details. – Martin Sleziak Aug 23 '12 at 16:01
@MartinSleziak Ok sir. Thank you. – Gomathi Aug 23 '12 at 16:05
up vote 1 down vote accepted

Look at the multiplication tables at the bottom of the page in your link, and try rewriting the second one with the columns and rows in the order $3,1,2$ instead of $1,2,3$. You should see something that looks almost identical to the table on the left, but with different symbols. Specifically, $a$ is replaced by $3$, $b$ by $1$ and $c$ by $2$. This is why the two are isomorphic - the two algebraic structures are the same, just the symbols for the elements are different. If you reshuffle the columns on the right-hand side in any other way, the tables won't match up properly, which is why other definitions of $g$ won't work.

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Thanks. But how to find the right combination? Is that by trial and error method? – Gomathi Aug 23 '12 at 16:02
Roughly, although you don't have to try every possibility. It's clear from the table that you have to either have $g(b)=1$ and $g(c)=2$ or $g(b)=2$ and $g(c)=3$, as $b,c$ and $2,3$ are the elements that give you the same answer no matter what you multiply by, and then checking how they multiply with $a$ and $1$ tells you which way round they have to be. – Matthew Pressland Aug 23 '12 at 16:10
Ok sir. I get it. Thank you so much for your help. – Gomathi Aug 23 '12 at 16:12
That's good, because that comment has very confusing typos in it! And I can't edit it now. I meant "...$g(b)=2$ and $g(c)=1$, as $b,c$ and $1,2$...", and "...checking how they multiply with $a$ and $3$..." – Matthew Pressland Aug 23 '12 at 16:17
Sir, I have another doubt. These two tables are of same size. But what if both are of different size. For eg. $+_4$ ([0],[1],[2],[3]) and B={0,1} with operation + ? – Gomathi Aug 23 '12 at 16:27

You will have to check to see if $g(a\ast b)=3\oplus 1 \\ g(b\ast a)=1\oplus 3 \\ g(a\ast c)=1\oplus 2 \\ g(c\ast a)=2\oplus 1 \\ g(b\ast c)=1\oplus 2 \\ g(c\ast b)=2\oplus 1 \\$

If all of these hold, then the map $g$ "preserves" the operation structures.

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May I add that it is usually a good idea to check whether the identity in $S$ maps to the identity in $P$ as the first step. – Alexander Gruber Aug 23 '12 at 15:55
Do you mean we have to check for all three values (1,2,3) for a,b and c? – Gomathi Aug 23 '12 at 15:57
@AlexanderGruber Based on the OP's question, we have no reason to expect that there is an identity, or even associativity... – rschwieb Aug 23 '12 at 16:00
@Gomathi I don't know why you would do that. I am working under the assumption you had two different sets $\{1,2,3\}$ and $\{a,b,c\}$ which are in no way related to each other. – rschwieb Aug 23 '12 at 16:01
I have edited my question. Kindly check it out. – Gomathi Aug 23 '12 at 16:03

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