### Disproof of Wall’s Conjecture

Institute: AIM     July 2013
During the June 2012 AIM workshop, “Cohomology bounds and growth rates” a counterexample was found to a group theory conjecture formulated by G. E. Wall in 1961.

A mathematical group is a set along with an operation that combines two elements to form a third element of the set and satisfies certain other axioms. A familiar example is that of the integers with the operation of addition. The concept of a group first appeared in the middle of the 19th century, although not in the abstract form that is found in modern textbooks. The French mathematician Galois defined a group in 1832, and the notion first appeared in print in 1846. Groups appear (often in disguise) in many areas of mathematics and science and understanding their basic structure is of fundamental importance in mathematics.

A subgroup is a group within a group, that is, a subset that with the same operation of the group also satisfies the group axioms. Some subgroups are contained in larger subgroups and the ones at the top of the chain, not contained in any other subgroup, are called maximal subgroups.

Fifty years ago Wall conjectured that the number of maximal subgroups of any finite group is less than the size of the group. Mathematical progress is rarely done quickly, but at times things come together in a serendipitous way, and that is what happened at the AIM workshop.

Wall himself had shown the conjecture to be true in some special cases in his 1961 paper. In 1986 Robert Guralnick, a workshop organizer, had also formulated a related relevant conjecture about a “cohomology bound.” Recently there was dramatic progress on the Guralnick conjecture, and as the workshop began it was thought that the time was ripe to prove the Wall conjecture. But instead, workshop participants found a counterexample.

First hint of a counterexample

On day three of the workshop Leonard Scott, also a workshop organizer, reported that one of his undergraduate students, Tim Sprowl, had run a computer programs to check some examples. It succeeded in obtaining some new answers, but crashed when it was 80 percent complete. Fortunately, taking part in the workshop was Frank Luebeck, currently in charge of the computer algebra project GAP and an expert in computational algebra, and he knew that he had a program on his desktop computer in Aachen that could complete the computation that had caused Sprowl’s program to crash and handle other cases as well.

Overnight he connected to his computer via the internet and carried out an amazing array of calculations that he presented to the workshop the next day. Bob Guralnick immediately realized that the results gave counterexamples to Wall’s conjecture, which came as such a surprise that some people in the audience took flash photos of Luebeck with their cell phones. Somewhat taken aback by all the attention, Luebeck spent the rest of the day thinking through the logic of his program to make sure it was right, and then Luebeck and Scott realized that Luebeck’s computations could be checked independently by Scott and Sprowl.

This was carried out several weeks later, when Sprowl got his program to run without crashing, and handled the next larger cases for the groups it was designed to treat. This indeed produced counterexamples to Wall’s conjecture and provided confirmation of a portion of Luebeck’s calculations. Scott and Sprowl exchanged e-mails with Luebeck confirming that their computations gave identical answers. The effect of Luebeck’s calculations at the conference had been, in Guralnick’s later words “revolutionary”. Perhaps more importantly, it became clear from results presented at the conference that the Guralnick and Wall conjectures had helped lead the subject of finite group cohomology to a genuinely higher level. It showed how much modern methods in conjunction with improvements in computer technology had put questions like the Wall conjecture within reach, whereas before they were beyond either theoretical or computational methods alone.

Biography of Wall

Gordon E. (Tim) Wall was born in Adelaide, Australia in 1925. He received his undergraduate degree at the Universities of Adelaide (BSc 1946) and and his Ph.D. in Cambridge in 1952. After this, he was an Assistant Lecturer in Mathematics at Manchester University from 1949 to 1951, a Lecturer in Mathematics, at Durham University, Newcastle-upon-Tyne from 1952 to 1956, and then a Senior Lecturer/ Reader in the Department of Pure Mathematics at the University of Sydney from 1956- to 1965. It was during this time that he formulated his important and fruitful conjecture. Wall became a professor at the University of Sydney in 1965 and an Emeritus Professor in 1991.

Wall made many fundamental contributions to the study of groups and algebras. The Wall polynomial, given this name by George Andrews in 1984, was studied by Wall on his work on conjugacy classes in classical groups.

His honors include becoming a Fellow of the Australian Academy of Science in 1971 and he was the President of the Australian Mathematical Society from 1970 to 1972.