Complete the group table

Ok the question says complete the group table to G = {a,b,c,d}
We are given this

x a b c d
a
b a
c a
d

I know that each element can occur only once per row/column and that one element needs to be the identity i just cant figure out which one im pretty sure if you figure out what element is the identity it wouldnt be to hard to fill in the rest im just unsure about how to figure it out.

Quote:

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Originally Posted by ChrisBickle

Ok the question says complete the group table to G = {a,b,c,d}
We are given this

x a b c d
a
b a
c a
d

I know that each element can occur only once per row/column and that one element needs to be the identity i just cant figure out which one im pretty sure if you figure out what element is the identity it wouldnt be to hard to fill in the rest im just unsure about how to figure it out.

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have you filled in anything else?

actually i just looked at it that its not right what i posted i should have b * b = a and a * c = a other than that im not sure where to go there is nothing (i dont think giving where the other a should go. I guess a * c = a so there is a possibility c is the identity?

Quote:

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Originally Posted by ChrisBickle

actually i just looked at it that its not right what i posted i should have b * b = a and a * c = a other than that im not sure where to go there is nothing (i dont think giving where the other a should go. I guess a * c = a so there is a possibility c is the identity?

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$\displaystyle ac=a\implies c=aa^{-1}=e$. So yes, $\displaystyle c=e$. From there you should be able to fill in quite a few squares.

Quote:

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Originally Posted by ChrisBickle

Ok the question says complete the group table to G = {a,b,c,d}
We are given this

x a b c d
a
b a
c a
d

I know that each element can occur only once per row/column and that one element needs to be the identity i just cant figure out which one im pretty sure if you figure out what element is the identity it wouldnt be to hard to fill in the rest im just unsure about how to figure it out.

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If I understand correctly, you have $\displaystyle a$ twice in the first column and this cannot be in a group's multiplication table...

Tonio

Quote:

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Originally Posted by tonio

If I understand correctly, you have $\displaystyle a$ twice in the first column and this cannot be in a group's multiplication table...

Tonio

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He corrected himself. I believe the table should should start off as,

x a b c d
a
b ..a
c a
d

This means that c is the identity, and so we have an element which isn't of order 2 in a group of order 4, namely b.

Anyway, the solution is got from here by filling in what the identity does, and then by pretending it is a simplified suduko...