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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)

Table

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?

Justin
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  • What is your question? – Irrational Person Feb 16 '15 at 22:39
  • Label as you like colors (or shades of gray) in the second picture here: http://mathworld.wolfram.com/DihedralGroupD4.html – MattAllegro Feb 16 '15 at 22:40
  • I edited the post, but the main question is in this part: 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? – Justin Feb 17 '15 at 00:35
  • Keep in mind you cannot assume $ab$ is equal to $ba$, so many of your entries are wrong even before you simplify them. For example you have $abab = a^2 b^2$ when in fact, $abab = abab$. – Josh B. Feb 17 '15 at 00:41
  • I cannot comment the previous answer. I want to clarify that, because D4 is a group, we know that the operation is associative, but we don't know if it is conmutative (in fact, it isn't and your errors come from here) Also, the sentence "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." holds because of the cancelative property that every group has because every element has its inverse element, and it is unique. Anyway, you can't even use associavity or cancelation if you have to prove that D4 is a group and, in this case, – FCardelle Feb 14 '19 at 13:25

1 Answers1

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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.

Josh B.
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