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Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ is at most $|G|^{100}$?

One can easily find that any subgroup is generated by at most $\log|G|$ elements thus the number of subgroups(in particular maximal subgroups) is at most $|G|^{\log|G|}$. Does it possible to improve this upper bound. For an abelian group the number of maximal subgroups is at most $|G|$ and in fact I do not know any example where the number of maximal subgroups is more than $|G|$.

I am almost sure that I am not the first who is asking this question I would like to know if the answer to this question is known or either this is a hard question.

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Another extended comment. The best bound on the number of subgroups is $|G|^{(1/4+o(1))\log_2 |G|}$ proved in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.913&rep=rep1&type=pdf by Borovik, Pyber and Shalev.

For the number of maximal solvable subgroups they get $|G|^c$ for some constant. They don't estimate c but conjecture it is 1.

Added. The survey László Pyber, Asymptotic results for simple groups and some applications. Groups and computation, II (New Brunswick, NJ, 1995), 309–327, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 28, Amer. Math. Soc., Providence, RI, 1997 claims there is a bound of the form $|G|^c$ for any group G on maximal subgroups but c is not described. It is conjectured c=1 will do.

Added. Jesse in the comments below cites a more recent and better bound of $c|G|^{3/2}$. I am making this CW to not get credit for his answer.

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    $\begingroup$ In the paper Martin W. Liebecka, Laszlo Pyberb, Aner Shalev: On a conjecture of G.E. Wall. Journal of Algebra, 317, no.1, p. 184-197 (2007) they give an upper bound of $c|G|^{3/2}$ for some absolute constant $c$. $\endgroup$ Aug 15, 2012 at 17:33
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    $\begingroup$ @Jesse, that should be an answer. $\endgroup$ Aug 15, 2012 at 17:54
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    $\begingroup$ There's no need to make this a CW. I learned of Wall's conjecture from Feng Xu who has considered generalizations for subfactors in a couple of papers. He asked a related question about this in April: mathoverflow.net/questions/94553/… $\endgroup$ Aug 15, 2012 at 18:59
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    $\begingroup$ NB There's a typo in the reference - that should be Martin Liebeck. $\endgroup$
    – HJRW
    Aug 15, 2012 at 20:51
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    $\begingroup$ There is a book called subgroup lattices. Google it. $\endgroup$ Aug 16, 2012 at 13:46
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The document I linked to above is sufficiently striking as to warrant an answer of its own. I hope it complements the community wiki above.

As mentioned above the relevant conjecture in this area is due to Wall:

Conjecture The number of maximal subgroups of a finite group $G$ is less than the order of $G$.

This has been the subject of much study with the landmark work (until recently) being the above-cited work of Liebeck, Pyber and Shalev. In addition to the result mentioned above they show that the conjecture is true if the group $G$ is simple, up to a finite number of exceptions.

Now a quote from the linked document is relevant:

This largely directed attention to composite groups, where Wall in his original paper had at least shown the conjecture to be true for finite solvable groups. The key remaining cases were known to be semidirect products of a vector space V with a nearly simple finite group G acting faithfully and irreducibly on it.

It turns out that in this case Wall's conjecture implies some bounds on the cohomology groups $H^1(G,V)$. And, as the document relates, examples have now been found which violate these bounds. In particular, Wall's conjecture does not hold.

In light of this development, the bound $C|G|^{3/2}$ mentioned above, also due to Liebeck, Pyber and Shalev, assumes greater importance. Although, as the linked document mentions, it is likely that the value $3/2$ can be reduced a great deal.

One final interesting quote:

A conjecture of Aschbacker and Guralnick, not made at the conference... would now rise to be the main conjecture in maximal subgroup theory. (The conjecture states that it is the number of conjugacy classes of maximal subgroups that is bounded, less than the number of conjugacy classes of elements in the group.)

Anyone interested should definitely read this document. Not only is it interesting mathematically, it's a very engaging account of how this recent breakthrough was achieved.

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  • $\begingroup$ Fantastic answer! I hadn't heard about this breakthrough, though like Jesse I've been interested in Wall's conjecture from the subfactor side. $\endgroup$ Dec 7, 2012 at 22:36
  • $\begingroup$ Is there a reference in which Aschbacker and Guralnick state their conjecture? $\endgroup$ May 30, 2016 at 15:23
  • $\begingroup$ @SebastienPalcoux, I don't have access to the following article but I believe it is where the conjecture is stated: M. Aschbacher, R. Guralnick Some applications of the first cohomology group J. Algebra, 90 (1984), pp. 446–460. $\endgroup$
    – Nick Gill
    Jun 1, 2016 at 9:02
  • $\begingroup$ Yes at the end of p447. Also previously, Conjecture B' p190 in Solvable generation of groups and Sylow subgroups of the lower central series, J. Algebra 77 (1982). 189-201. You should be interested in new results by Feng Xu about this conjecture: Symmetries of subfactors motivated by Aschbacher–Guralnick conjecture, Adv. in Math., 289 (2016) 345–361. $\endgroup$ Jun 1, 2016 at 9:25

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