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#### conscipost

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- Jan 26, 2012

- 39

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- Jan 26, 2012

- 39

- Jan 26, 2012

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Basically, this questions asks whether 3 elements: a, b, e with the following properties:

a^2=a

b^2=b

e^2=e

ea=ae=a

eb=be=b

ab=ba=e

form a group.

My answer is no, because it is not isomorphic to $Z_3$.

Simpler reason:

(ab)b=eb=b

a(bb)=ab=e

Associative property is not satisfied.

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- Jan 26, 2012

- 39

That's true. Thanks for pointing that out.Basically, this questions asks whether 3 elements: a, b, e with the following properties:

a^2=a

b^2=b

e^2=e

ea=ae=a

eb=be=b

ab=ba=e

form a group.

My answer is no, because it is not isomorphic to $Z_3$.

Simpler reason:

(ab)b=eb=b

a(bb)=ab=e

Associative property is not satisfied.

I suppose at the least it is an interesting counter example.

If concentration was considered I can imagine this situation working though.

So, b+b=2b and a+(2b)=b. This would leave the 3 element structure it has now, and I suppose would be isomorphic to (Z,+) where multiples of a are negative integers and multiples of b positive integers.

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