Geodesic
Self-normalizing nilpotent subgroups of the full linear group over a~finite field
Zapiski Nauchnykh Seminarov POMI, Algebraic numbers and finite groups, Tome 86 (1979), pp. 34-39

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It has been proved (Ref. Zh. Mat., 1977, 4A170) that in the full linear group $GL(n,q)$, $n=2,3$, over a finite field of $q$ elements, $q$ odd or $q=2$, the only self-normalizing nilpotent subgroups are the normalizers of Sylow 2-subgroups and that for even $q>2$ there are no such subgroups. In the present note it is deduced from results of D. A. Suprunenko and R. F. Apatenok (Ref. Zh. Mat., 1960, 13586; 1962, 9A150) that this is true for any $n$. 
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     author = {N. A. Vavilov},
     title = {Self-normalizing nilpotent subgroups of the full linear group over a~finite field},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {34--39},
     publisher = {mathdoc},
     volume = {86},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_86_a4/}
}
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N. A. Vavilov. Self-normalizing nilpotent subgroups of the full linear group over a~finite field. Zapiski Nauchnykh Seminarov POMI, Algebraic numbers and finite groups, Tome 86 (1979), pp. 34-39. http://geodesic.mathdoc.fr/item/ZNSL_1979_86_a4/