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Other than showing if a group is abelian and the order of the group, what else does a Cayley table indicate?

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It 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. – Arturo Magidin Oct 13 '11 at 19:53
In 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. – Arturo Magidin Oct 13 '11 at 19:58
Arturo 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). – Bill Cook Oct 13 '11 at 20:00
You can compute the center easily by looking at the table. – Mariano Suárez-Alvarez Oct 13 '11 at 20:28
And 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! – Nicky Hekster Oct 13 '11 at 21:02

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