I'm currently trying to learn abstract algebra myself, and the following is a quote from the book I am using, "A set of equations, involving only the generators and their inverses, is called a set of defining equations for $G$ if these equations completely determine the multiplication table of $G$."
Then the book proceeds to give an example: "Let $G$ be the group $\{e, a, b, b^{2}, ab, ab^{2} \}$ whose generators $a$ and $b$ satisfy the equations $a^{2} = e$, $b^{3} = e$, and $ba = ab^{2}$." And claims that the three equations determine the multiplication table of $G$.
So I worked out the multiplication table and displayed it below.
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When they say, "completely determine the multiplication table of $G$," does that mean the product of two elements can be simplified to another element? For example, $(ab^{2})(ab^{2}) = ab(ba)bb = ab(ab^{2})b^{2} = abab(b^{3}) = a(ba)b = a(ab^{2})b = aab^{3} = e.$
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I also don't see how inverses are used in determining the multiplication table in this case. I've only used substitution in this case. Can someone explain why inverses might be important?
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How did the author know that only 3 equations were enough to determine the multiplication table? And why did he choose those equations?
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Also what is the significance of determining a multiplication table for elements of a group?
Multiplication Table of G
___________________________________________
| e a b b^2 ab ab^2 |
.-------+-------------------------------------------+
| e | e a b b^2 ab ab^2 |
| a | a e ab ab^2 b b^2 |
| b | b ab^2 b^2 e a ab |
| b^2 | b^2 ab e b ab^2 a |
| ab | ab b^2 ab^2 a e b |
| ab^2 | ab^2 b a ab b^2 e |
'-------+-------------------------------------------'
Best Answer
Fun fact to know: in 1992 Ales Drápal proved that if two finite groups agree on 89% on their multiplication tables, the groups must be isomorphic! He conjectured that the same holds true if the tables agree on 75% of their entries. The conjecture has not been proved yet. See also Groups St. Andrews 2001 at Oxford, featuring the paper of Drápal On the distance of 2-groups and 3-groups.