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

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.

• I asked this on the group-pub about a year ago, and I received the following email from Mark Lewis (forwarded from Ron Solomon): Richard and I HOPE to complete Book 7 by the end of 2012. Richard, Inna Capdeboscq and I have a lot of work completed on Book 8 as well. We might be able to complete it in one to 1 1/2 years after the completion of Book 7. That still leaves Books 9, 10, and 11. (The current best estimate is 11 total.) Gernot Stroth has a first draft of Book 11 written. Some work has been done on Book 9, but I would not hazard a guess on when that will see the light of day. Nov 30 '12 at 1:27
• @Steve, would you care to post this as an answer? Dec 3 '12 at 1:48
• unfortunately, no. Book 7 should be out by now, but I haven't heard anything about an upcoming release. Feb 15 '14 at 3:53
• Scientific American had a popular article on this topic in its July 2015 issue (Vol 313, No. 1). German version in Spektrum der Wissenschaft, März 2016. Oct 4 '16 at 14:29
• @FriederLadisch Thanks! The article is Researchers Race to Rescue the Enormous Theorem before Its Giant Proof Vanishes by Stephen Ornes (doi:10.1038/scientificamerican0715-68). I quote: "Solomon and Lyons are finishing the seventh book this summer, and a small band of mathematicians have already made inroads into the eighth and ninth. Solomon estimates that the streamlined proof will eventually take up 10 or 11 volumes, which means that just more than half of the revised proof has been published." So I guess that book 7 should be done, but presumably submitted somewhere? Oct 5 '16 at 4:52

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.)

• Book 7 is supposed to come out in March 2018. bookstore.ams.org/surv-40-7 Jan 31 '18 at 18:40
• @TimothyChow this would be better as an answer, to bump the question for increased visibility. EDIT: actually, I'll edit my answer instead Mar 27 '18 at 0:10
• @TimothyChow thanks for the edit to my answer! I'm looking for material that discusses this and future volumes. If you know of any, can you point me to it? Feb 25 at 6:17
• @DavidRoberts The only thing I know about is the progress report by Ron Solomon in the June/July 2018 issue of the Notices of the AMS. Feb 25 at 14:15

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.