Here is an excerpt.
Now consider [the multiplication table of] the group with three distinct elements:

. The only products which we must determine explicitly are

,

,

, and

since all other products involve the unit

. The product

cannot equal

or

, since that would imply that either

or

, respectively. Thus,

. The Cancellation Theorem then
requires that

[The book misprints this equation.],

, and

.
My problem is that I can't justify to myself the last sentence. Please help explain.
P.S.
Emmy Noether is German.
