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In finite group theory, $p$-groups or simple groups can be considered as building blocks of all the groups. What is known about these families of groups is that

  1. The classification of simple groups is done.

  2. The classification of $p$-groups is much difficult problem.

Due to classification of simple groups, many problems from different areas of mathematics have been solved; there is a book Atlas of Finite Simple Groups: Ten years on, which perhaps contains some work with this thought.

On the other hand, although classification of $p$-groups is difficult, many researchers are focused on classifying $p$-groups having a specific property. For example, such work is collected in five big volumes (each $\geq$ 600 page, multiple of thousands of problems in total), only devoted to finite $p$-groups by Berkovich and Janko, in which, the authors state in preface that their aim is classification w.r.t. some properties.

I wondered many times, but didn't get any definite answer to following question while discussing with some people working on $p$-groups. I thought it may not be good to post this question on group-pub-forum to de-motivate some group-theoriests. The question is

As long as classification of some types of $p$-groups is concerned, has its application appeared in some other branch of mathematics (or at least, in algebra, or at least in group theory itself!)?

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  • $\begingroup$ Using number theory techniques you can study zeta functions of groups which have really nice properties. So optimistically you could say there are $p$-groups that can help you study some meromorphic function of possible interest. $\endgroup$ – Ali Caglayan Mar 24 '18 at 14:31
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    $\begingroup$ "$p$-groups or simple groups" are considered as building blocks? certainly not in the same way. For simple groups there's a pretty clear statement (Jordan-Hölder). $p$-groups play an essentially role, first because they have the nice behavior predicted by Sylow theorems, and are instrumental in the classification of simple groups, etc. But the parallel is somewhat irrelevant, I think (you may think of $p$-groups as building blocks if you like, but not in the same way). $\endgroup$ – YCor Mar 24 '18 at 17:04
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    $\begingroup$ The classification of finite simple groups is an extraordinarily difficult, yet solved problem. The classification of $p$-groups is not just "difficult", it's something which is admittedly not the right question, in the same way that we don't try any longer to "classify" algebraic numbers or prime numbers. At least, I think nobody thinks of a "workable list" as that of finite simple groups (by the way, the classification of finite simple groups is only a classification modulo the "classification" of prime numbers!!). $\endgroup$ – YCor Mar 24 '18 at 17:08
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Understanding the structure of various classes of $p$-groups already plays a substantial role in the classification of finite simple groups. To give (only a few of many possible), understanding $p$-groups in which all characteristic Abelian subgroups are cyclic ( so called $p$-groups of symplectic type), which were classified by P. Hall, is important in local analysis used for classification of simple groups. Understanding $p$-groups in which the graph with vertices elementary Abelian subgroups of order $p^{2}$ is disconnected, where two such subgroups are joined if they commute (elementwise) is relevant for the proof of the odd order theorem, and in signalizer functor theory elsewhere. Understanding $2$-groups of sectional rank at most $4$ is useful for CFSG. There are many other examples, and maybe more will occur during the ongoing revision projects for CFSG.

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Let me speak to part of your question: What after it? We can not reasonably expect to classify the $p^{2n^3/27 +O(n^{8/3})}$ groups of order $p^n$ for large $n$. We can classify $p$-groups with particular properties focusing on those with applications to other problems (as Geoff and Alireza mention). Alternatively, we can focus on understanding small and highly symmetric examples. Certainly in graph theory small and highly symmetric examples have played an important role e.g. $s$-arc transitive graphs (with $2\leqslant s\leqslant 7$) have applications to incidence geometry and abstract polytopes. This philosophy was behind our paper Maximal linear groups induced on the Frattini quotient of a p-group. Indeed, one could argue that most large $p$-groups with small automorphism groups are uninteresting in the same way that most large graphs with few symmetries are uninteresting.

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The answer is positive: since one must give a classification of at least one types of finite $p$-groups which I suggest to consider the classification of finite $p$-groups having a maximal subgroup which is cyclic; and next one must show us an application in group theory as it is mentioned above.

For the latter one may look at the following book and then count the number of usage of the above classification (Theorem 1.2, page 22) to see how many application the above classification may have at least. Note that "Application of an Application of the classification" must be considered as "an Application of the classification".

Berkovich, Yakov Groups of prime power order. Vol. 1. With a foreword by Zvonimir Janko. De Gruyter Expositions in Mathematics, 46. Walter de Gruyter GmbH & Co. KG, Berlin, 2008. xx+512 pp. ISBN: 978-3-11-020418-6

For example, the classification of non-abelian finite 2-groups $G$ with $|G:G'|=4$ (Proposition 1.6, page 26) follows from the classification of groups with a cyclic maximal subgroups.

Another application of finite $p$-groups which is every time one use is the classification of groups of order at most $p^3$. This has certainly many applications in group theory, one can certainly find such applications in the following book:

Leedham-Green, C. R.; McKay, S. The structure of groups of prime power order. London Mathematical Society Monographs. New Series, 27. Oxford Science Publications. Oxford University Press, Oxford, 2002. xii+334 pp. ISBN: 0-19-853548-1

Let me finish my answer by saying that the ultimate aim of every branch of mathematics is to classify all objects which are studied therein; however the latter aim may not be possible, where the impossibility must be defined among the branch.

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    $\begingroup$ I don't agree with the statement that "the ultimate aim of every branch of mathematics is to classify all objects which are studied therein." $\endgroup$ – Andy Putman Jan 31 '17 at 19:11
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    $\begingroup$ Andy: Then the ultimate aims of mathematics fall into at least two classes! $\endgroup$ – Geoffrey Irving Jan 31 '17 at 19:24
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    $\begingroup$ Those examples of application seem to be a bit circular: Classification of p-groups are used to classify more p-groups, or to understand certain clases of p-groups... Are you aware of any application that are not by themselves directly about p-groups? $\endgroup$ – Max Horn Feb 1 '17 at 11:36

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