Geodesic
On certain near-domains and sharply $2$-transitive groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 277-283

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We find sufficient conditions under which a near-domain is a near-field and a $2$-transitive group has a normal regular abelian subgroup. If a sharply $2$-transitive group $T$ ($\mathrm{Char}\,T\ne2$) contains a Frobenius group with involution such that its complement contains a subgroup of order $>2$ that is normal in the stabilizer of a point, then $T$ has a regular abelian normal subgroup (Theorem 1). If, in a near-domain of odd characteristic, there is a near-field containing a multiplicative subgroup of order $>2$ that is normal in a multiplicative group of the near-domain, then the near-domain is a near-field (Theorem 2). This result also holds in the case when the local nilpotent radical of the stabilizer of a point contains a $2$-subgroup of order $\geq16$ and the characteristic is congruent to 1 modulo 16 (Theorem 3).
Mots-clés : group, Frobenius group.
Keywords: near-field, near-domain 
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     title = {On certain near-domains and sharply $2$-transitive groups},
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A. I. Sozutov; E. B. Durakov; E. V. Bugaeva. On certain near-domains and sharply $2$-transitive groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 277-283. http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a23/