Large linear groups of nilpotence class two
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Archiv der Mathematik
Large linear groups of nilpotence class two Hangyang Meng
Abstract. Let V be a non-trivial finite-dimensional vector space over a finite field F of characteristic p and let G be an irreducible subgroup of GL(V ) having nilpotence class at most two. We prove that if |G| > |V |/2, then G is cyclic, or |V | = 32 or 52 . This is a refinement of Glauberman’s result for the tight bound of linear groups of nilpotence class two. Mathematics Subject Classification. 20C15, 20D10. Keywords. Linear groups, Modules, Nilpotence class two.
1. Introduction. Let V be a non-trivial finite-dimensional vector space over a finite field F of characteristic p, p a prime. In [1, Proposition 1], Glauberman proved that if G is a p -subgroup of GL(V ) having nilpotence class at most two, the order of G is at most |V | − 1. Note that this bound is the best. For example, there indeed exists a cyclic subgroup of GL(V ) with order |V | − 1, and if |V | = 32 , GL(V ) has subgroups isomorphic to Q8 and D8 . Our first result shows that the linear groups whose orders reach the best bound only belong to the two cases mentioned above. Theorem 1. Let V be a non-trivial finite-dimensional vector space over a finite field F of characteristic p, p a prime, and let G be a p -subgroup of GL(V ) having nilpotence class at most two with order |V | − 1. Then (a) G is cyclic; or (b) |V | = 32 and G is isomorphic to Q8 or D8 . Note that if G is a linear subgroup of order less than |V |, then every proper subgroup of G has order at most |V |/2. In order to find the linear groups with large order, we study the irreducible linear group such that |G| > |V |/2 in Theorem 2. This research is sponsored by the Shanghai Sailing Program (20YF1413400) and the Young Scientists Fund of NSFC (12001359).
H. Meng
Arch. Math.
Recall that a linear group is called irreducible if it is irreducible on its natural module. Given two linear groups Gi ≤ GL(Vi ), i = 1, 2, these two actions (or modules) of Gi on Vi are called similar if there exist a groupisomorphism α : G1 → G2 and a linear isomorphism φ : V1 → V2 such that (vg)φ = (vφ)(gα)
∀g ∈ G1 , v ∈ V1 .
Theorem 2. Let V be a non-trivial finite-dimensional vector space over a finite field F of characteristic p, p a prime, and let G be an irreducible subgroup of GL(V ) with nilpotence class at most two. Then |G| ≤ |V |/2 unless one of the following cases holds: (a) G is cyclic and |G| = |V | − 1; (b) |V | = 32 and G ∼ = D8 or Q8 ; (c) |V | = 52 and the action of G on V is similar to the action of the following subgroup of order 24 in GL(2, 5) on GF (5) ⊕ GF (5): 02 20 1 0 01 20 1 0 , , or , , . 10 02 0 −1 10 02 0 −1 We can extend Theorem 2 to the general case. Corollary 3. Let V be a non-trivial finite-dimensional vector space over a finite field F of characteristic p, p a prime, and let G be a non-abelian p -subgroup of GL(V ) having nilpotence class two such that |G| > |V |/2. Then G/Z(G) is an elementary abelian 2-group and p = 3 or 5. 2. Proof of Theorem 2. Most of the not
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