r/3 are multiplicative inverses.
In 1940, an algebraist named Graham Higman made a daring conjecture in his doctoral thesis: The worst of this cancellation weirdness, he proposed, will only happen if the group that is used to construct the group algebra contains elements for which some power equals 1, as with
r in the example above. In all other group algebras, he posited, while elements with just a single term, like 7
a or 8
r + 2 or 3
r â 5
s can never have multiplicative inverses. Since elements with multiplicative inverses are called units, Higmanâs hypothesis came to be known as the unit conjecture.
Over the next few decades, Irving Kaplansky, one of the leading mathematicians of the 20th century, popularized this conjecture along with two other group algebra conjectures called the zero divisor and idempotent conjectures; the three came to be known as the Kaplansky conjectures. Collectively, the three conjectures posit that group algebras are not too radically