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In this paper, we first study biplanes \(\mathcal {D}\) with parameters (v, k, 2), where the block size \(k\in \{13,16\}\). These are the smallest parameter values for which a classification is not available. We show that if \(k=13\), then either \(\mathcal {D}\) is the Aschbacher biplane or its dual, or \(\mathbf {Aut}(\mathcal {D})\) is a subgrou...
Contexts in source publication
Context 1
... a Sylow p-subgroup of Aut(D) with p ∈ {5, 7, 11, 13} is cyclic of order at most p. Moreover, the cycle type of a p-element x ∈ Aut(D) on P and on a fixed block (if such exists) is given in Table 4. ...Context 2
... first that p = 11. Then since p is coprime to 121, x fixes at least one point, and so by Lemma 2, x fixes also a block, say B. Table 4, and hence every p-element has exactly s fixed points in B where s is listed in the second column Table 4. Now let P be a Sylow p-subgroup of Aut(D) where p ∈ {5, 7, 13}. Then P acts faithfully on B. Since p 2 does not divide k − s, it follows that P has an orbit ∆ of length p in [B]. ...Context 3
... first that p = 11. Then since p is coprime to 121, x fixes at least one point, and so by Lemma 2, x fixes also a block, say B. Table 4, and hence every p-element has exactly s fixed points in B where s is listed in the second column Table 4. Now let P be a Sylow p-subgroup of Aut(D) where p ∈ {5, 7, 13}. ...Context 4
... P α = 1, and hence |P | = |P : P α | = |∆| = p. Therefore, P is cyclic of order p and the cycle types of the non-identity elements of P on P and on B are as in the fourth and fifth columns of Table 4, respectively. ...Similar publications
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Citations
The existence of a biplane with parameters (121,16,2) is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a possible biplane D of order 14 divides 27⋅32⋅5⋅7⋅11⋅13. In this paper we show that such a biplane does not have an automorphism of order 11 or 13, and thereby establish that |Aut(D)| divides 27⋅32⋅5⋅7. Further, we study a possible action of an automorphism group of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters (121,16,2).
