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Math Help - Write the table of G...

  1. #1
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    Write 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...
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by jzellt View Post
    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...
    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,

    \begin{array}{ccccccccc}\:& e & a & b & b^2 & b^3 & ab & ab^2 & ab^3\\ e&\:&\:&\:&\:&\:&\:&\:&\:\\<br />
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 e, and go across right-multiplying by every element in the top row. Then go to the second top element in the column, a, and right multiply this by every element in the top row, so your top two rows will look like,

    \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\\<br />
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, a^2 and 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, 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).
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