Proof of Abelian Property - a * b * c = e

[jeff1evesque]

Statement:
37. Let G be a group and suppose that $$a * b * c = e$$ for $$a, b, c \in G.$$

Problem:
Show that $$b * c * a = e.$$

Thought Process
Can we assume that our binary operator $$*$$ is abelian. Thus, $$a * b * c = a * (b * c) = (b * c) * a = b * c * a.$$

Thanks,

JL

 

[winter85]

Hi jeff1evesque,

You can't assume the group is abelian when the question doesn't say it is abelian. Actually, the result is true even if the group is not abelian.
The associative law is helpful. can you state in plain english what a*(b*c) = e means?

 

[jeff1evesque]

Hi jeff1evesque,

You can't assume the group is abelian when the question doesn't say it is abelian. Actually, the result is true even if the group is not abelian.
The associative law is helpful. can you state in plain english what a*(b*c) = e means?

It means that the binary operation on the three elements a, b, c produces the identity. Many possibilities can be true, depending on what the binary operator does. One possibility could be that the result of some operation between any of the two elements (a, b, c) yields to some element say r. When we take the operation of r with the third element we get the identity. There may however, be other possibilities (since the operation itself is undefined) which produces the same identity element $$e$$.

 

[Billy Bob]

Statement:
37. Let G be a group and suppose that $$a * b * c = e$$ for $$a, b, c \in G.$$

Problem:
Show that $$b * c * a = e.$$

Here's an idea that is (I think) different from winter85's, even though that way would work too.

First multiply both sides of $$a * b * c = e$$ by a cleverly chosen element. "Repeat" as necessary, but not necessarily with the same element.

To receive full credit for your write-up, just take care that you left-multiply both sides, or right-multiply both, but don't left-multiply one side and right-multiply the other.

 

[winter85]

It means that the binary operation on the three elements a, b, c produces the identity. Many possibilities can be true, depending on what the binary operator does. One possibility could be that the result of some operation between any of the two elements (a, b, c) yields to some element say r. When we take the operation of r with the third element we get the identity. There may however, be other possibilities (since the operation itself is undefined) which produces the same identity element $$e$$.

if you let q = (b*c), the you have a*q = e.. what is q called in this case, with respect to a? what are its properties?