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Let S be a 2-subgroup of the K-automorphism group Aut(X) of an algebraic curve X of genus g(X) defined over an algebraically closed field K of characteristic 2. It is known that S may be quite large compared to the classical Hurwitz bound 84(g(X)−1). However, if S fixes no point, then the size of S is smaller than or equal to 4(g(X)−1). In this paper, we investigate algebraic curves X with a 2-subgroup S of Aut(X) having the following properties: (I) |S|≥8 and |S|>2(g(X)−1), (II) S fixes no point on X. Theorem 1.2 shows that X is a general curve and that either |S|=4(g(X)−1), or |S|=2g(X)+2, or, for every involution u∈Z(S), the quotient curve X/〈u〉 inherits the above properties, that is, it has genus ≥2, and its automorphism group S/〈u〉 still has properties (I) and (II). In the first two cases, S is completely determined. We also give examples illustrating our results. In particular, for every g=2h+1≥9, we exhibit a (general bielliptic) curve X of genus g whose K-automorphism group has a dihedral 2-subgroup S of order 4(g−1) that fixes no point in X.

@article{ 11391_1344516, author = { Giulietti Massimo and Korchmáros Gábor }, title = {Large 2-groups of automorphisms of algebraic curves over a field of characteristic 2}, year = {2015}, journal = {JOURNAL OF ALGEBRA}, volume = {427}, abstract = {Let S be a 2-subgroup of the K-automorphism group Aut(X) of an algebraic curve X of genus g(X) defined over an algebraically closed field K of characteristic 2. It is known that S may be quite large compared to the classical Hurwitz bound 84(g(X)−1). However, if S fixes no point, then the size of S is smaller than or equal to 4(g(X)−1). In this paper, we investigate algebraic curves X with a 2-subgroup S of Aut(X) having the following properties: (I) |S|≥8 and |S|>2(g(X)−1), (II) S fixes no point on X. Theorem 1.2 shows that X is a general curve and that either |S|=4(g(X)−1), or |S|=2g(X)+2, or, for every involution u∈Z(S), the quotient curve X/〈u〉 inherits the above properties, that is, it has genus ≥2, and its automorphism group S/〈u〉 still has properties (I) and (II). In the first two cases, S is completely determined. We also give examples illustrating our results. In particular, for every g=2h+1≥9, we exhibit a (general bielliptic) curve X of genus g whose K-automorphism group has a dihedral 2-subgroup S of order 4(g−1) that fixes no point in X.}, keywords = {Algebraic curves, Positive characteristic, Automorphism groups}, url = {http://www.sciencedirect.com/science/article/pii/S0021869314007182}, doi = {10.1016/j.jalgebra.2014.12.019}, pages = {264--294} }

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