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. (G is called dihedral group D4)
However, there are some elements that are not in the group like B$^2$ so I have to rewrite it but I do not know how to re-write it. I know for A$^2$=e since that is given but how do i apply it to other elements. For example, A$^2$B$^2$, it is not a column or row.Therefore it is not in group G and must be rewritten, how can i rewrite it?
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$\begingroup$First remember that the group operation here is concatenation, which is to say, $a\ast b=ab$. Some of your entries are incorrect even before you simplify them. Like I said in the comments, $ab\ast ab=abab$, not $a^2b^2$.
Next, use the relations to simplify the entries.
For example, $a\ast a=a^2$ but from the relations $a^2=e$, so $a\ast a=e$.
Also, $ab\ast b^3=ab^4$ and from the relations $b^4=e$, so $ab\ast b^3=a$.
Some of these simplifications may take a few steps: $ab\ast ab=abab=a(ba)b=a(a b^3)b=a^2\ast b^4=e$
Also to help check your work each row and column of the table should have one and only one of the elements as a product.
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