After the conjecture was posed in the 1970s, dozens of mathematicians tried their hand at proving it. They made partial progress — and in the process they learned a great deal about groups, which are abstract objects that describe the various symmetries of a mathematical system. But a full proof seemed out of reach.
Then Späth came along. Now, 20 years after she first learned about the problem and more than a decade after she met Cabanes, the two mathematicians have finally completed the proof.
When the couple announced their result, their colleagues were in awe. “I wanted there to be parades,” said Persi Diaconis of Stanford University. “Years of hard, hard, hard work, and she did it, they did it.”
The Power of Primes
The McKay conjecture began with the observation of a strange coincidence.
John McKay — described by one friend as “brilliant, soft-spoken, and charmingly disorganized” — was known for his ability to spot numerical patterns in unexpected places. The Concordia University mathematician is perhaps most famous for his “monstrous moonshine” conjecture, which was proved in 1992 and established a deep connection between the so-called monster group and a special function from number theory.
Before his death a few years ago, McKay unearthed lots of other important connections, too, many involving groups. A group is a set of elements combined with a rule for how those elements relate to one another. It can be thought of as a collection of symmetries — transformations that leave a shape, a function or some other mathematical object unchanged in specific ways. For all their abstraction, groups are immensely useful, and they play a central role in mathematics.
In 1972, McKay was focused on finite groups — groups that have a finite number of elements. He observed that in many cases, you can deduce important information about a finite group by looking at a very small subset of its elements. In particular, McKay looked at elements that form a special, smaller group — called a Sylow normalizer — inside the original group.
Imagine you have a group with 72 elements. This alone doesn’t tell you much: There are 50 different groups of that size. But 72 can also be written as a product of prime numbers, 2 $latex \times$ 2 $latex \times$ 2 $latex \times$ 3 $latex \times$ 3 — that is, as 23 $latex \times$ 32. (Generally, the more distinct primes you need to describe the size of your group, the more complicated your group is.) You can decompose your group into smaller subgroups based on these primes. In this case, for instance, you could look at subgroups with eight (23) elements and subgroups with nine (32) elements. By studying those subgroups, you can learn more about the structure of your overall group — what other building blocks the group is composed of, for instance.
Now take one of those subgroups and add a few particular elements to it to create a special subgroup, the Sylow normalizer. In your 72-element group, you can build a different Sylow normalizer for each eight-element and nine-element subgroup — these are the 2-Sylow normalizers and 3-Sylow normalizers, respectively.
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