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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 For example, DANIEL GORENSTEIN, in his paper "ON A THEOREM OF PHILIP HALL", mentioned the unpublished lecture notes of Philip Hall.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 Since the notes do not be published, I can not find way to see them.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?

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up vote 24 down vote accepted

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 is located online here:

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."

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There's a copy of the Edmonton notes in the Cambridge maths library; I consulted it a couple of years ago. – Ben Green Mar 19 '12 at 11:03
According to the Edmonton notes are still in print – Michael Giudici Mar 19 '12 at 11:44
Appreciate Prof. Humphreys gives me a helpful answer. I have read the note in "The collected works of Philip Hall". The notes mentioned in the footnote of the above paper by Gorenstein, which I am very interesting, seems different from the note "The Edmonton notes on nilpotent groups" or notes in "The collected works of Philip Hall". I think that the reference [20] in "solvability of groups of odd order" by Feit and Thompson, which I really want to see, is the same as the notes mentioned in the paper by Gorenstein. – Wei Zhou Mar 19 '12 at 12:46
There are also "digital" copies of the revised edition floating around... – Steve D Mar 19 '12 at 16:39
Taking it as an example, let's talk about the theorem at the end of answer given by Prof. Humphreys. This theorem is interesting and important. There are some generation about this theorem. As I know, beside the proof given by Gorenstein, Berkovich give another proof. But the two proofs are not natural to me. Since Hall is so important in modern p-group, so I hope the see the original proof of Hall. The theorem may have some background, which is als interesting for me. – Wei Zhou Mar 20 '12 at 5:06

Apparently the original notes have been published online by Washington University here

I must add that I'm an admirer of Hall's math, and its an emotion to read his handwritten notes.

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It's great! I can't believe that I see these notes finally. Thank you very much! – Wei Zhou Aug 1 '15 at 8:36

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