Math couple solves major group theory problem after 20 years.
Sylow normalizers, like the subgroups they’re built out of, can tell mathematicians a lot about the original group. But McKay hypothesized that this connection was far stronger than anyone had imagined. It wasn’t just that a Sylow normalizer could give insights into a finite group’s overall structure. He asserted that if mathematicians wanted to compute a crucial quantity that would help them characterize their group, they’d just have to look at one of a particular set of Sylow normalizers: The Sylow normalizer would be characterized by the exact same number.
This quantity counts the number of “representations” of a certain type — ways you can rewrite elements of the group using arrays of numbers called matrices. Such a tally might seem arbitrary, but it gives mathematicians a sense of how the group’s elements relate to each other, and it is involved in calculations of other important properties.
