Geodesic
Finite Projective Planes with Affine Subplanes
Canadian mathematical bulletin, Tome 7 (1964) no. 4, pp. 549-559

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A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2). 
Ostrom, T. G.; Sherk, F. A. Finite Projective Planes with Affine Subplanes. Canadian mathematical bulletin, Tome 7 (1964) no. 4, pp. 549-559. doi: 10.4153/CMB-1964-051-8
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     author = {Ostrom, T. G. and Sherk, F. A.},
     title = {Finite {Projective} {Planes} with {Affine} {Subplanes}},
     journal = {Canadian mathematical bulletin},
     pages = {549--559},
     year = {1964},
     volume = {7},
     number = {4},
     doi = {10.4153/CMB-1964-051-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1964-051-8/}
}
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