Polarities in translation planes

Ganley, Michael J.

doi:10.1007/bf00147384

Cited by 35 publications

(25 citation statements)

References 9 publications

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“…A proof of this result may be found in [3]. Here, we prove the corresponding result for semifield planes of even order, and in fact we prove rather more, namely that if such a unique involution exists, then it is the only non-trivial collineation of/7 which fixes the subplane pointwise.…”

mentioning

confidence: 76%

“…Proof. It is well-known that if S and S' have the same elements, then multiplication in S' is given by x.y= (xR~ l)(yL~ l) for some non-zero a, rueS, where Ra denotes right multiplication by a, and L m denotes left multiplication by m. The multiplication on the right-hand side of the above equation is merely the multiplication in S (see, for instance, [3] and the references contained therein).…”

Section: Lemma 5 F Is Solvable If and Only If R Is Solvablementioning

confidence: 99%

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Baer involutions in semifield planes of even order

Ganley

1974

Geom Dedicata

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mentioning

confidence: 76%

Section: Lemma 5 F Is Solvable If and Only If R Is Solvablementioning

confidence: 99%

Baer involutions in semifield planes of even order

Ganley

1974

Geom Dedicata

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“…Consider the projective plane Π(K) coordinatized by a Dickson semifield K. By a result of Ganley [12,Theorem 5], any projective plane coordinatized by a finite commutative semifield which has a non-trivial involutory automorphism admits a unitary polarity. In particular, since the Dickson semifield K admits an involutory automorphism α :…”

Section: Dickson Semifield Planes and Dickson-ganley Unitalsmentioning

confidence: 99%

“…Here we turn our attention to a class of polar unitals embedded in the Dickson semifield planes. In [12], Ganley showed that the projective plane Π(K) defined over a Dickson semifield K admits a unitary polarity which thus defines a polar unital U. The unital U = U(σ) is parametrized by a non-identity field automorphism σ.…”

Section: Introductionmentioning

confidence: 99%

Non-classical polar unitals in finite Dickson semifield planes

et al. 2013

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Abstract. We give three proofs, two intrinsic and one extrinsic, that every Dickson-Ganley unital U(σ), parametrized by a field automorphism σ, is non-classical if σ is not the identity, extending a result of Ganley's (Math Z 128:34-42, 1972); we prove that U(σ1) is isomorphic to U(σ2) if and only if σ1 = σ2 or σ1 = σ −1 2 ; and we determine the (design) automorphism group of U(σ) as the collineation subgroup of the ambient Dickson semifield plane stabilizing the unital. This contains as a special case the corresponding result of O'Nan's (J Algebra 20:495-511, 1965) on the classical unital. Mathematics Subject Classification (2000). 51A35, 05B05, 17A35, 51A10, 51A45.

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“…Then A(S • ) can be coordinatized by a commutative semifield by [Ga,Theorem 3], and hence the desired k exists by [Ga,Theorem 4]. (ii) We will prove later in Theorem 4.12 that the autotopism group of A * is isomorphic to a subgroup of Aut(F ), so that the hypotheses of Theorem 3.30 hold.…”

Section: Semifield Planes and Codes 911mentioning

confidence: 99%

Symplectic semifield planes and ℤ₄–linear codes

Kantor

Williams

2003

Trans. Amer. Math. Soc.

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Abstract. There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z 4 -linear Kerdock and Preparata codes on the other. These interrelationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non-isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z 4 -linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2.We also obtain large numbers of "boring" affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes "boring" in the sense that each has as small an automorphism group as possible.The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.

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Polarities in translation planes

Cited by 35 publications

References 9 publications

Baer involutions in semifield planes of even order

Baer involutions in semifield planes of even order

Non-classical polar unitals in finite Dickson semifield planes

Symplectic semifield planes and ℤ₄–linear codes

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