A finite groups of $\mathrm{GL}\_n(\mathbb C)$ of exponent $m$ necessarily have order $C$ verifying $C\leqslant m^n$ and $n! m^n$ divides $C$, but this condition is not sufficient, for instance $\mathrm{GL}\_3(\mathbb C)$ has no subgroup of exponent $7$ and order $42$. 


So can we determine fully the possible orders of the subgroups of $\mathrm{GL}\_n(\mathbb C)$ of exponent $m$?


~~~~~~~~~~~~OLD QUESTION - Which was obvious, sorry and thanks Pete ~~~~~~~~~~~~~~~~


After some computation and search it seems that finite subgroups of $\mathrm{GL}\_3(\mathbb C)$ of exponent $5$ have order $5$, $5^2$ or $5^3$.
Similarly, finite subgroups of $\mathrm{GL}\_3(\mathbb C)$ of exponent $7$ have order $7$, $7^2$ or $7^3$.


So, for $n >2$ and a prime $p >2$, is it true that finite subgroups of $\mathrm{GL}\_n(\mathbb C)$ of exponent $p$ necessarily have order of the form $p^k$? I apologize in advance if this is obvious and I missed it.



Finite subgroups of GL_n(C)
The paper:


Herzog, Marcel; Praeger, Cheryl E. "On the order of linear groups of fixed finite exponent."
J. Algebra 43 (1976), no. 1, 216–220.
[MR424960](http://www.ams.org/mathscinet-getitem?mr=424960)
[DOI:10.1016/0021-8693(76)90156-3](http://dx.doi.org/10.1016/0021-8693(76)90156-3)


contains the important bound, if G ≤ GL(n,F) where F has characteristic coprime to |G|, then |G| ≤ exp(G)n. Obviously these bounds can be obtained over large enough F (containing exp(G) roots of unity), as the diagonal subgroup generated by *e*th roots of unity has exponent e and order en.


The exponent of a finite group divides the order of the group: the exponent of a group is the product of the exponents of its Sylow subgroups, and a p-group always contains an element whose order is equal to the exponent of the group, so by Lagrange the exponent divides the order. Also, every prime dividing the order of the group divides the exponent of the group, by Cauchy's theorem.


In particular, the possible orders of finite groups of exponent p, p a prime, that are contained in GL(n,C) are exactly p1, p2, …, pn. The elementary abelian subgroups generated by diagonal matrices whose entries are *p*th roots of unity shows the existence, and Herzog and Praeger eliminate all other orders. Note that when p is large, these are all possible anyways, so that the theorem is probably only interesting for small p.


For instance, GL(3,C) contains no non-abelian group of exponent 5.


