About unpublished lecture notes of Philip Hall

2012-03-19T07:21:27Z

Dear all: When I study group theory, I find there are some mysterious things. For example, DANIEL GORENSTEIN, in his paper "ON A THEOREM OF PHILIP HALL", mentioned the unpublished lecture notes of Philip Hall. Many other famous group theorists also confirmed that these notes are important to their work. Since the notes do not be published, I can not find way to see them. But I am very curious about what's in these notes? Are there any theory about groups we don't know or do not appear in published books or papers?

Answer by Jim Humphreys for About unpublished lecture notes of Philip Hall

2012-03-19T10:57:47Z

It may help to have some explicit bibliographic references, though some items are by now out of print and may be difficult to locate even through libraries. First, the 1966 paper by Gorenstein can be located online:

Gorenstein, Daniel. On a theorem of Philip Hall. Pacific J. Math. 19 (1966), 77–80.

Presumably he is referring to the 1957 lectures by Hall, which circulated in small numbers and were later published at QMC in London (where I may even have seen them, since I was lecturing there in 1969):

Hall, Philip. The Edmonton notes on nilpotent groups. Queen Mary College Mathematics Notes. Mathematics Department, Queen Mary College, London 1969 iii+76 pp.

These typewritten notes have largely disappeared from view by now, but fortunately a version got included in the 1988 volume of collected works:

Hall, Philip. The collected works of Philip Hall. Compiled and with a preface by K. W. Gruenberg and J. E. Roseblade. With an obituary by Roseblade. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1988. xii+776 pp. ISBN 0-19-853254-7.

Here is a brief excerpt from the review in Mathematical Reviews:

"The works appear in chronological order and are reproduced in their original format apart from the Edmonton notes on nilpotent groups (based on Hall’s lectures given at the summer seminar of the Canadian Mathematical Congress held in 1957) which are a corrected reprint of the third edition (Queen Mary College, 1979)."

The book is also hard to find now outside some libraries, but does exist. As reviews indicate, much but not all of the material in Hall's original lecture notes has appeared in various research papers. Good luck.

ADDED: I'm sorry to have gone off on the wrong trail in response to the question raised here. Certainly Philip Hall's influence on other group theorists was not limited to his formally published work, though I suspect the informal stuff left out of his collected works might be covered indirectly through the papers he inspired other people to write. Though not many people directly involved in that era remain, it would for example be interesting to know what John Thompson's recollections are.

Perhaps it may inspire some finite group theorists to suggest better answers if I quote more precisely the "theorem of Philip Hall" which Gorenstein revisited from the viewpoint of the 1963 Feit-Thompson paper:

"If $P$ is a $p$ -group with no noncyclic characteristic abelian subgroups, then $P$ is the central product of subgroups $P_1$ and $P_2$ , where $P_1$ is extra-special and either $P_2$ is cyclic or $p=2$ and $P_2$ is dihedral, generalized quaternion, or semi-dihedral."

About unpublished lecture notes of Philip Hall - MathOverflow

About unpublished lecture notes of Philip Hall

Answer by Jim Humphreys for About unpublished lecture notes of Philip Hall