Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?

Question

According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher and Smith in 2004, and one by Aschbacher, Lyons, Smith and Solomon in 2011. However, this latter book is really just the second part of an outline of the proof, the first part of which was written by Gorenstein in the 80s (the reason for the delay is, of course, that the quasithin case hadn't actually been settled at the time of the announcement of completion). Hence the last update on the second-generation proof is really 2005.

With the recent formal proof in Coq of the Odd-order Theorem, it would be good to know where the traditional proof is up to.

EDIT 6 August 2013: Any news as to the completion of that seventh volume as mentioned in the comments?

EDIT 29 September 2016 Just a bump to this question in case people know more about where the progress is at. Books 7 and 8 should probably have made some progress since I asked this originally.

Answer 1

There is an interesting review by Ron Solomon of a paper in this area, which has been featured on the Beyond Reviews blog. In particular, he outlines the broad tactics that people are using in CFSG II, and some of the content that will be going into volume 7.

Also, Inna Capdeboscq apparently gave an outline of volume 8, or at least a chunk of it, at the Asymptotic Group Theory conference in Budapest. This was mentioned by Peter Cameron on his blog, sadly with no detail! If anyone can get a whiff of what she said, I would be grateful.

EDIT 15 October 2016 I emailed the group-pub mailing list and was told second-hand that Ron Solomon 'has hopes' volume 7 will be submitted next year.

EDIT 27 March 2018 Thanks to Timothy Chow in a comment on another answer, here is the link to the published version of Volume 7. So now the countdown to Volume 8 starts...

EDIT 22 June 2018 Even better news: Volume 8

...is near completion and promised to the AMS by August 2018. The completion of Volume 8 will be a significant mathematical milestone in our work. (source)

Also (from the same article):

We anticipate that there will be twelve volumes in the complete series [GLS], which we hope to complete by 2023.

Considerable work has been done on this problem [the bicharacteristic case], originally by Gorenstein and Lyons, and more recently by Inna Capdeboscq, Lyons, and me. We anticipate that this will be the principal content of Volume 9 [GLS], coauthored with Capdeboscq.

When p is odd, there is a major 600-page manuscript by Gernot Stroth treating groups with a strongly p-embedded subgroup, which will appear in the [GLS] series, probably in Volume 11. There are also substantial drafts by Richard Foote, Gorenstein, and Lyons for a companion volume (Volume 10?), which together with Stroth’s volume will complete the p-Uniqueness Case.

It would be wonderful to complete our series by 2023, the sixtieth anniversary of the publication of the Odd Order Theorem. Given the state of Volumes 8, 9, 10, and 11, the achievement of this goal depends most heavily on the completion of the e(G) = 3 problem. It is a worthy goal.

EDIT Mar 2019 Volume 8 has been published. The page listing the available volumes, along with links to more details is here.

The summary of this volume is as follows:

This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach a major milestone in our series—the completion of the proof of the following theorem:

Theorem O: Let G be a finite simple group of odd type, all of whose proper simple sections are known simple groups. Then either G is an alternating group or G is a finite group of Lie type defined over a field of odd order or G is one of six sporadic simple groups.

Put another way, Theorem O asserts that any minimal counterexample to the classification of the finite simple groups must be of even type. The work of Aschbacher and Smith shows that a minimal counterexample is not of quasithin even type, while this volume shows that a minimal counterexample cannot be of generic even type, modulo the treatment of certain intermediate configurations of even type which will be ruled out in the next volume of our series.

EDIT February 2021 Volume 9 has now been published. From the preface:

This book contains a complete proof of Theorem $\mathcal{C}_5$, which covers the “bicharacteristic” subcase of the $e(G) \ge 4$ problem. The outcome of this theorem is that $G$ is isomorphic to one of the six sporadic groups for which $e(G)\ge 4$, or one of six groups of Lie type which exhibit both characteristic 2-like and characteristic 3-like properties. Finally, in Chapter 7, we begin the proof of Theorem $\mathcal{C}_6$ and its generalization Theorem $\mathcal{C}^∗_6$, which cover the “$p$-intermediate” case. $\ldots$ In the preceding book in this series, we had promised complete proofs of Theorems $\mathcal{C}_6$ and $\mathcal{C}^∗_6$ in this book, but because of space considerations, we postpone the completion of those theorems to the next volume.

EDIT September 2021

In response to a question from Hugo de Garis, Ron Solomon sent the following email in January 2021:

Vol. 9 is already submitted, accepted and scheduled for publication. It should be published early this year. As for the rest, my best guess now is that there will in fact be 4 further volumes, not 3. A roughly 800 pages manuscript on the Uniqueness Theorem has been completed by Gernot Stroth. With some additional material, it will fill 2 further volumes. This could probably be readied for publication by a year from now. However, our team (Inna Capdeboscq, Richard Lyons, Chris Parker and myself) are currently focussing on the remaining work to be done for the other two volumes. It is difficult to estimate how long this will take. With luck we might have a first draft completed this calendar year, but it might take longer. It is safe to say that the remaining volumes will not all be published before 2023. I hope it is also safe to say that they will all be published no later than 2025.

(Emphasis added)

EDIT 29 Dec 2021

Richard Lyons maintains an erratum for the whole published second generation CFSG on this page: https://sites.math.rutgers.edu/~lyons/cfsg/

EDIT 05 Apr 2022

In response to a further question from de Garis (see the page linked above), Solomon wrote (in March 2022):

We have been working on the theorems for both Volumes 10 and 11. Just in the past few weeks, we have decided to concentrate on the completion of Volume 10. This is proceeding very well and we should be able to submit Volume 10 for publication this year, I believe. I fear that I may have been a bit overoptimistic in predicting the completion of all the volumes by the end of 2024.

EDIT 09 Mar 2023

From a 23 January 2023 article about Inna Capdeboscq (emphasis added):

The expected length of the Generation-2 proof is of about 5,000 pages published in 12 volumes. At this moment Volumes 1 through 9 are published. Inna has been involved in the Generation-2 project for several years, providing small contributions to Volume 6 and 7. Inna co-authored the recently published Volume 9 and is currently in a process of completing Volume 10.

I don't know how this estimate of 12 volumes sits with Solomon's email from January 2021 (see the Sept '21 edit) saying there would be 4 more volumes after vol 9 was done. And though Stroth's future contribution is mentioned in the short article, I don't think this count of 12 includes his manuscript mentioned above.

EDIT 24 June 2023

I emailed Richard Lyons to double check how things are going given the hopeful progress on volume 10, mentioned above. He replied (and he and Ron Solomon gave permission to relay this):

Volume 10 has been submitted for publication.

I have received Stroth's final manuscript for the Uniqueness Case, which we plan to make the final volume.

[Ron] Solomon and I are currently working on Volume 11, the penultimate volume (provided that it fits into one volume .. it is not clear at this time whether it will or not). This will complete the proof of Theorem C_4 (the last of the seven in the Classification Grid) and begin the treatment of the Uniqueness Case for groups of even type, to mesh with Stroth's work.

So it seems Stroth will contribute one volume, to go at the end, and we will have volume 11 (and maybe vol 12) of the main series before that. So 11+2+1 (or 12+2+1) volumes in total. [edit: the +2 is the Aschbacher–Smith work, the +1 is Stroth]

EDIT 9 October 2023

Commenter colt_browning points out below that Volume 10 is due for publication 26th December, and is now available for preorder: https://bookstore.ams.org/surv-40-10. The title is The Classification of the Finite Simple Groups, Number 10: Part V, Chapters 9–17: Theorem $C_6$ and Theorem $C^*_4$, Case A, with listed authors Capdeboscq, Gorenstein, Lyons and Solomon, and it's 570 pages long.

This book is the tenth in a series of volumes whose aim is to provide a complete proof of the classification theorem for the finite simple groups based on a fairly short and clearly enumerated set of background results. Specifically, this book completes our identification of the simple groups of bicharacteristic type begun in the ninth volume of the series (see Mathematical Surveys and Monographs, Volume 40.9). This is a fascinating set of simple groups which have properties in common with matrix groups (or, more generally, groups of Lie type) defined both over fields of characteristic 2 and over fields of characteristic 3. This set includes 11 of the celebrated 26 sporadic simple groups along with several of their large simple subgroups. Together with SURV/40.9, this volume provides the first unified treatment of this class of simple groups.

Total length of volumes 1–10 is 4511 pages, and Aschbacher and Smith's two volumes fill 1320 pages. Maybe another 1000–1500 pages to go? There's an old manuscript of Stroth from the late 90s that seems to cover the "uniqueness case" (first listed article on this page), which is what his volume will cover. That's 244 pages, but it's not clear how it relates to the draft of what will become the last volume of the published second generation proof.

Answer 2

With respect to the second generation proof you can get an answer `from the horse's mouth' if you like: Ron Solomon gave an update on the program at BIRS recently and a video of his talk is here.

He starts the talk by comparing its progress to `the receding of the glaciers'! In reality, though, they've made very significant headway into the later volumes. (In particular he mentions, around 9:30, that volume 7 is in preparation, and it is mainly this volume that he's discussing in the talk.)