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Is anything known about the function $P(n)$ where,

$$P(n)=|\{ m\leq n :\text{There is a perfect group of order } m\}|.$$

Like asymptotics or a good upper bound?

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  • $\begingroup$ Its a cool question but it lacks context $\endgroup$
    – Jakobian
    Mar 4 at 4:05
  • $\begingroup$ "Enumeration of finite groups" (referenced here) might address your question. (I don't have a copy to hand.) $\endgroup$ Mar 4 at 4:21
  • $\begingroup$ I wonder whether almost all perfect numbers are the orders of ${\rm PSL}(2,q)$ or ${\rm SL}(2,q)$ for a prime power $q$. In other words numbers of the form $q(q^2-1)$, or $q(q^2-1)/2$ when $q$ is odd. $\endgroup$
    – Derek Holt
    Mar 5 at 9:55
  • $\begingroup$ ... or possibly even numbers of the form $p(p^2-1)/2$ with $p$ an odd prime. $\endgroup$
    – Derek Holt
    Mar 5 at 10:04
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    $\begingroup$ @verret I would certainly guess that to be true, but I have no idea how you might go about proving it. $\endgroup$
    – Derek Holt
    Mar 6 at 8:11

2 Answers 2

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This doesn't answer the question that you asked, but if we let ${\rm Perf}(\le n)$ be the number of isomorphism classes of perfect groups of order up to $n$, then we have the bounds $$n^{l(n)^2/108-cl(n)} \le {\rm Perf}(\le n) \le n^{l(n)^2/48 + l(n)},$$ where $c$ is a constant and $l(n)= \log_2(n)$. This is proved in this paper: "Enumerating Perfect Groups" by D. F. Holt, J. London Math, Soc. 39 (1989), 67-78.

Since there are very large numbers of perfect groups of certain orders, such as $2^n.60$, this doesn't help much with estimating your function $P(n)$.

Note that the crude estimate $n^{O((\log n) ^2)}$ is the same as for the growth rate of all finite groups. But the constant in the exponent is $2/27$ for all groups, and something between $1/108$ and $1/48$ for perfect groups.

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I'm not sure what is known about this function, but you might be interested to know that it has (basically) been computed for small values of $n$.

GAP comes with a library of finite perfect groups, including the function SizesPerfectGroups which returns a list of all integers between $1$ and $2 \cdot 10^6$ which occur as the orders of perfect groups. That list is

[ 1, 60, 120, 168, 336, 360, 504, 660, 720, 960, 1080, 1092, 1320, 1344, 1920, 2160, 2184, 2448, 2520, 2688, 3000, 3420, 3600, 3840, 4080, 4860, 4896, 5040, 5376, 5616, 5760, 6048, 6072, 6840, 7200, 7500, 7560, 7680, 7800, 7920, 9720, 9828, 10080, 10752, 11520, 12144, 12180, 14400, 14520, 14580, 14880, 15000, 15120, 15360, 15600, 16464, 17280, 19656, 20160, 21504, 21600, 23040, 24360, 25308, 25920, 28224, 29120, 29160, 29760, 30240, 30720, 32256, 32736, 34440, 34560, 37500, 39600, 39732, 40320, 43008, 43200, 43320, 43740, 46080, 48000, 50616, 51840, 51888, 56448, 57600, 57624, 58240, 58320, 58800, 60480, 61440, 62400, 64512, 64800, 65520, 68880, 69120, 74412, 75000, 77760, 79200, 79464, 79860, 80640, 84672, 86016, 86400, 87480, 92160, 95040, 96000, 100920, 102660, 103776, 110880, 112896, 113460, 115200, 115248, 115320, 116480, 117600, 120000, 120960, 122472, 122880, 126000, 129024, 129600, 131040, 131712, 138240, 144060, 146880, 148824, 150348, 151200, 151632, 155520, 158400, 159720, 160380, 161280, 169344, 172032, 174960, 175560, 178920, 180000, 181440, 183456, 184320, 187500, 190080, 192000, 194472, 201720, 205200, 205320, 216000, 221760, 223608, 225792, 226920, 230400, 232320, 233280, 237600, 240000, 241920, 243000, 244800, 244944, 245760, 246480, 254016, 258048, 259200, 262080, 262440, 263424, 265680, 276480, 285852, 288120, 291600, 293760, 300696, 302400, 311040, 320760, 322560, 332640, 336960, 344064, 345600, 352440, 357840, 360000, 362880, 363000, 364320, 366912, 367416, 368640, 369096, 372000, 375000, 378000, 384000, 387072, 388800, 388944, 393120, 393660, 410400, 411264, 411540, 417720, 423360, 432000, 435600, 443520, 446520, 447216, 450000, 451584, 453600, 456288, 460800, 460992, 464640, 466560, 468000, 475200, 480000, 483840, 489600, 491520, 492960, 504000, 515100, 516096, 518400, 524880, 531360, 544320, 546312, 550368, 552960, 571704, 574560, 583200, 587520, 589680, 600000, 604800, 604920, 607500, 612468, 622080, 626688, 633600, 645120, 647460, 665280, 673920, 675840, 677376, 685440, 688128, 691200, 693120, 699840, 704880, 712800, 720720, 721392, 725760, 728640, 729000, 730800, 733824, 734832, 737280, 748920, 768000, 774144, 777600, 786240, 787320, 806736, 816480, 820800, 822528, 823080, 846720, 864000, 871200, 874800, 878460, 881280, 885720, 887040, 892800, 900000, 903168, 907200, 912576, 921600, 921984, 929280, 933120, 936000, 937500, 943488, 950400, 950520, 960000, 962280, 967680, 976500, 979200, 979776, 983040, 987840 ]

From this you can easily compute your function $P(n)$ for $n \leq 2 \cdot 10^6$. Here's a quick plot from Mathematica:

Plot of P(n)

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