Other than showing if a group is abelian and the order of the group, what else does a Cayley table indicate?
What does a Cayley table tell?
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$\begingroup$
abstract-algebra
group-theory
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4It tells you the result of every multiplication of two elements in the group... Do you mean, is there some other useful and nontrivial information that can be gleaned out of the table just by staring at it for a while? Very little: that's why we don't usually study groups by staring at their Cayley tables. – 2011-10-13
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0Yes, I filled out the Cayley table, but I need to say what else it shows. (how it's useful) – 2011-10-13
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7In a sense, the Cayley table tells you *everything you could possibly want* about a group, because it tells you exactly what the operation on the group is. You can use it to compute any product, you can use it to figure out inverses, etc. The issue is that most of these things are not particularly *easy* to do with the table (as opposed to checking commutativity, which can be done "at a glance"). "How it's useful" is a rather vague question to ask. – 2011-10-13
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2Arturo is right. Generally it's hard to say a whole lot about a generic group (other than abelian or not) just by looking at its Cayley table. It's even hard to tell if two groups are isomorphic just by looking at their Cayley tables side by side (the elements may not be in the "right" order to see the isomorphism). On the other hand, if you organize elements according to some subgroup and its cosets, you can use the table to "see" the corresponding quotient group (or see that the quotient does not exist). – 2011-10-13
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0Okay, I see. I'm looking at the Cayley table of Q8. I have found that the group is not abelian. I am to find one other significant fact about Q8 by looking at its Cayley table. – 2011-10-13
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2You can compute the center easily by looking at the table. – 2011-10-13
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0Oh! Thanks everyone! – 2011-10-13
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1And if the rows and columns are arranged properly you can read off subgroups! For example in your quaternion group (most of the time represented as the set {1,i,j,k,-1,-i,-j,-k}) if you start the row with 1, -1, i, -i and do the same for the column, you notice the subgroup generated by i. And also the center {1,-1} as indicated by Mariano! – 2011-10-13
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1I might add that if your the Cayley table is not too big also the order of an element can be computed rather easily. – 2011-10-13