I was wondering if there is a reference studying groups with order $m^k$ where $m,k$ are integers and $m$ is not supposed to be a prime, as a generalization of $p$-groups?
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$\begingroup$ I guess the right generalization would be to study classes of finite groups in the generated by a group variety under taking extensions. A closer case would be to consider all groups with prime divisors in some given finite set, e.g., all groups of order dividing $6^n$ for some $n$. Indeed the class you're suggesting is not stable under taking subgroups/quotients when $m$ is not a prime power. $\endgroup$– YCorCommented Jul 19, 2019 at 8:05
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1$\begingroup$ Seems unlikely to admit a fairly general approach, because you'd be studying all groups of order $m$ of necessity (as your class would include the groups $G^k$ for any $|G|=m$), and that can get very ad hoc and complex if $m$ isn't very nice. And just about every property that makes $p$-groups study-able in (more-or-less) their entirety are all dependent on the primality. Are there particular things you're hoping to know about such groups? $\endgroup$– zibadawa timmyCommented Jul 19, 2019 at 8:15
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$\begingroup$ An example of a reference is Burnside's $p^aq^b$ Theorem (Wikipedia link). $\endgroup$– YCorCommented Jul 19, 2019 at 8:57
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3$\begingroup$ A better generalization is given by finite nilpotent groups. Every finite $p$-group is nilpotent and by a well- known theorem, each finite nilpotent group is isomorphic to the direct product of its Sylow subgroups. $\endgroup$– Victor ProtsakCommented Jul 19, 2019 at 9:14
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