Let G be the group {e, a, b, b^2, b^3, ab, ab^2, ab^3} whose generators satisfy:

a^2 = e, b^4 = e, ba = ab^3. Write the table of G.

Can someone show what the table looks like and how to get it? Thanks for any help...

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- Sep 28th 2010, 08:21 PMjzelltWrite the table of G...
Let G be the group {e, a, b, b^2, b^3, ab, ab^2, ab^3} whose generators satisfy:

a^2 = e, b^4 = e, ba = ab^3. Write the table of G.

Can someone show what the table looks like and how to get it? Thanks for any help... - Sep 29th 2010, 12:46 AMSwlabr
I won't show you what it looks like, but I will show you how to get it.

Start by writing the elements across the way and down, as you probably have in your notes,

$\displaystyle \begin{array}{ccccccccc}\:& e & a & b & b^2 & b^3 & ab & ab^2 & ab^3\\ e&\:&\:&\:&\:&\:&\:&\:&\:\\

a&\:&\:&\:&\:&\:&\:&\:&\:\\b&\:&\:&\:&\:&\:&\:&\:& \:\\b^2&\:&\:&\:&\:&\:&\:&\:&\:\\b^3&\:&\:&\:&\:&\ :&\:&\:&\:\\ab&\:&\:&\:&\:&\:&\:&\:&\:\\ab^2&\:&\: &\:&\:&\:&\:&\:&\:\\ab^3\end{array}$

Then, go to the top element in the column, it's the $\displaystyle e$, and go across right-multiplying by every element in the top row. Then go to the second top element in the column, $\displaystyle a$, and right multiply this by every element in the top row, so your top two rows will look like,

$\displaystyle \begin{array}{ccccccccc}\:& e & a & b & b^2 & b^3 & ab & ab^2 & ab^3\\ e&e&a&b&b^2&b^3&ab&ab^2&ab^3\\

a&a&a^2&ab&ab^2&ab^3&a^2b&a^2b^2&a^2b^3\\b&\:&\:&\ :&\:&\:&\:&\:&\:\\b^2&\:&\:&\:&\:&\:&\:&\:&\:\\b^3 &\:&\:&\:&\:&\:&\:&\:&\:\\ab&\:&\:&\:&\:&\:&\:&\:& \:\\ab^2&\:&\:&\:&\:&\:&\:&\:&\:\\ab^3\end{array}$

Then repeat with all the other rows until the table is completely filled. You will notice that it is filled mostly with `words' outwith the given set (for example, $\displaystyle a^2$ and $\displaystyle b^4$ are not in the set). So your next stage is to apply the relations you are give which the generators satisfy. That is, re-write your table looking at each word in turn and using these relations to turn the element into an element from your set.

So, for example, $\displaystyle a^2b=eb=e$.

You should end up with a table containing only elements from the given set, and which is NOT symmetric along the diagonal. Also, every element should appear precisely once in every row and column (think suduko).