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Tuesday, April 26, 2022

How big will the classification of finite simple groups be?

Gorenstein, Lyons, and Solomon (and friends) have been writing a consolidated proof for the classification theorem for finite simple groups...since 1994. In fact, it's still a work in progress. How big will it be?

Thus far, nine volumes have been published:

Volume NumberFirst page numberBibliography Page numberNumber of Pages
122159137
214216202
3 18 420 402
4 18350332
5 14476462
6 14532518
7 12352340
8 14498484
9 12530518

At present, there is an expected additional 4 volumes coming (according to David Roberts's answer on Mathoverflow). Performing a linear regression on the number of pages in terms of the volume number, we can make a prediction. Here's the R code doing this:

df <- data.frame(vol = 1:9, 
      start=c(22,14,18,18,14,14,12,14,12), 
      bib=c(159,216,420,350,476,532,352,498,530));
df$pgs <- df$bib - df$start;
lm0 <- lm(pgs ~ vol, data=df);
summary(lm0);

# Call:
# lm(formula = pgs ~ vol, data = df)
# 
# Residuals:
#     Min      1Q  Median      3Q     Max 
# -118.29  -53.62  -14.82   84.78  105.84 
# 
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)   
# (Intercept)   174.56      62.68   2.785   0.0271 * 
# vol            40.53      11.14   3.639   0.0083 **
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Residual standard error: 86.27 on 7 degrees of freedom
# Multiple R-squared:  0.6542,	Adjusted R-squared:  0.6048 
# F-statistic: 13.24 on 1 and 7 DF,  p-value: 0.008296

predict(lm0, data.frame(vol=c(10,11,12,13)))
#        1        2        3        4 
# 579.8889 620.4222 660.9556 701.4889 

sum(df$pgs)
# [1] 3395

sum(predict(lm0, data.frame(vol=c(10,11,12,13)))) + sum(df$pgs)
# [1] 5957.756

Tl;dr: the classification of finite simple groups is expected to be around 5957 pages, but could ostensibly reach up to 8243 pages.

Addendum

We can improve the estimate by dropping the constant term, and estimating the "additional pages" compared to the first volume:

df <- data.frame(vol = 1:9, 
      start=c(22,14,18,18,14,14,12,14,12), 
      bib=c(159,216,420,350,476,532,352,498,530));
df$pgs <- df$bib - df$start;
df2 <- data.frame(vol = 1:9,
      pgs=df$pgs - df$pgs[1]);
lm0 <- lm(pgs ~ 0 + vol, data=df2);
summary(lm0)

# Call:
# lm(formula = pgs ~ 0 + vol, data = df2)
# 
# Residuals:
#     Min      1Q  Median      3Q     Max 
# -122.24  -37.17  -24.70   92.68  125.61 
# 
# Coefficients:
#     Estimate Std. Error t value Pr(>|t|)    
# vol   46.463      4.901    9.48 1.26e-05 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Residual standard error: 82.75 on 8 degrees of freedom
# Multiple R-squared:  0.9183,	Adjusted R-squared:  0.908 
# F-statistic: 89.86 on 1 and 8 DF,  p-value: 1.263e-05

predict(lm0, data.frame(vol=c(10,11,12,13))) + df$pgs[1]
#        1        2        3        4 
# 601.6316 648.0947 694.5579 741.0211 

sum(predict(lm0, data.frame(vol=c(10,11,12,13))) + 137) + sum(df$pgs)
# [1] 6080.305

This gives us a better fit (adjusted R-squared of 0.908) with a better error estimate (with 95% probability, it will be 6080 pages give or take 40 pages).

Tl;dr2: the classification of finite simple groups can be expected to be around 6080±40 pages long.

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