On the Normal Structure of NonCommutative Association Schemes of Rank 6

Abstract

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory. One of these generalizations is the observation that, similar to groups, association schemes of finite order are commutative if they have at most five elements and not necessarily commutative if they have six elements. While there is (up to isomorphism) only one noncommutative group of order 6, there are infinitely many pairwise non-isomorphic noncommutative association schemes of finite order with six elements. (Each finite projective plane provides such a scheme, and non-isomorphic projective planes yield non-isomorphic schemes.) In this note, we investigate noncommutative schemes of finite order with six elements which have a symmetric normal closed subset with three elements. We take advantage of the classification of the finite simple groups.

Key Words:

• Association schemes
• Graph automorphisms
• Semilinear maps of finite fields

2010 Mathematics Subject Classification:

• 05E30
• 20B20
• 05E18
• 15A04
• 05C50

ACKNOWLEDGEMENT

The author gratefully acknowledges the kind hospitality and the comfortable working environment which he experienced at the Max-Planck-Institut für Mathematik at Bonn.

Notes

¹The definition of an association scheme as well as all other scheme theoretic notation and terminology mentioned in this introduction will be given shortly after the statement of the results.

²We notice that Proposition 1.1 can also be obtained from [4, Lemma 3.4.1] together with [7, Theorem 4.1.3(iii)]. However, in contrast to the proof given in [4], the argument that we present in this note does not require representation theory of association schemes.

³Coxeter schemes are schemes which are defined on the set of all maximal flags of a building (in the sense of Tits). If a Coxeter scheme has six elements, the corresponding building is a projective plane. Coxeter schemes have been investigated in general in the last two chapters of [7] and in [8].

⁴Semidirect products of association schemes are defined in [7, Section 7.3].

⁵We observe that the definition of a scheme that we use in the present article differs from the more general definition given in [7]. In fact, the schemes that we consider in the present article are exactly the schemes in the sense of [7] that are defined on finite sets.

⁶Each thin scheme can be viewed in a natural way as a finite group; cf. [7, Theorem 5.5.1]. Moreover, by [7, Theorem 5.5.2], each finite group can be viewed as a thin scheme. From [7, Theorem 5.5.3] and [7, Theorem 5.5.4] one also knows that both of these correspondences are inverses of each other. Thus, our definition of a scheme generalizes the notion of a finite group.

⁷Here we deviate from the notation in [3].

⁸In fact, from Lemma 3.2(i) together with Corollary 4.5, one obtains even 7 ≤ l.