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Not sure how to make tables, but:

For a binary operation $*$, and set $\{e,a,b,c\}$, in the Cayley table, $a*a$ can be filled with either the identity or an element different from both $e$ and $a$. If in the table the place for $a*a$ is filled with $e$, then the rest of the table can be filled out in two different ways.

What are the two different ways?

Thanks

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2 Answers

Two days ago I computed all Cayley tables for the Gorup $(G,*)$ with $|G|=4$ and came up with 4 distinct Cayley tables (!):

For the abelian group of the itegers modulo $4$: $(\mathbb{Z}/(4),+_4)$.
$1.$ $$\begin{array}{|c|c|c|c|c|} \hline +_4 & e & a & b & c\\ \hline e & e & a & b & c \\ \hline a & a & b & c & e \\ \hline b & b & c & e & a \\ \hline c & c & e & a & b \\ \hline \end{array}$$ For the Group formed by the $\mathbb{C}$-roots of $z^4=1$: $(\xi_4,\cdot_\mathbb{C})$
$2.$ with $\xi_4=\{ 1,i,-i,-1 \}$
$$\begin{array}{|c|c|c|c|c|} \hline \cdot_\mathbb{C} & e & a & b & c\\ \hline e & e & a & b & c \\ \hline a & a & c & e & b \\ \hline b & b & e & c & a \\ \hline c & c & b & a & e \\ \hline \end{array}$$ $3.$ with $\xi_4=\{1,-1,i,-i \}$ $$\begin{array}{|c|c|c|c|c|} \hline \cdot_\mathbb{C} & e & a & b & c\\ \hline e & e & a & b & c \\ \hline a & a & e & c & b \\ \hline b & b & c & a & e \\ \hline c & c & c & e & a \\ \hline \end{array}$$ And for the Klein-$4$-Group: $(\mathbb{Z}_2\times\mathbb{Z}_2,+_{\mathbb{Z}_2\times\mathbb{Z}_2})$
$4.$ $$\begin{array}{|c|c|c|c|c|} \hline +_{\mathbb{Z}_2\times\mathbb{Z}_2} & e & a & b & c\\ \hline e & e & a & b & c \\ \hline a & a & e & c & b \\ \hline b & b & c & e & a \\ \hline c & c & b & a & e \\ \hline \end{array}$$ The last two are the ones you are looking for.

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The problem with "counting Cayley tables" is that they mostly only come up at the beginning of one's study of group theory, whereas it takes some group-theoretic sophistication to understand how and what to count, especially when two Cayley tables are to be regarded as distinct. In order to avoid a lot of tables which are permutations of each other, I would have expected the convention that whenever a product of two group elements is an element that hasn't yet appeared above and to the left of it in the table, one should name the new element with the next available letter in the alphabet. –  Pete L. Clark May 15 '13 at 23:31
    
Your second Cayley table violates this by setting $a*a = c$, when we could equally well have put $a*a = b$. The difference, notice, is only that we have interchanged the last two rows and columns of the table. Why do we want to count these as distinct? –  Pete L. Clark May 15 '13 at 23:32
    
I get what you mean, I'm still missing the knowledge to classify by isomorphism these tables. is that where you are going? –  dktr.k1 May 15 '13 at 23:55
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Hint: If $a * a = e$, then the values of $b * a$, $c * a$, $a*b$ and $a*c$ are determined (since each row/column contains each group element exactly once). What are the remaining possibilities for $b*b$? Can they all be extended to give a Cayley table for a group?

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The tables I get are (e a b c by e a b c) e a b c a e c b b c e a c b a e and e a b c a e c b b c a e c b e a for $a*a$ =$e$. For $a*a=b$, I get e a b c a b c e b c e a c e a b. Sorry that is such a mess to look at. I don't understand how any 2 of these are isomorphic to each other. –  achacttn Oct 7 '12 at 20:45
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