A bijective mapping φ: G → G defined on a finite group G is complete if the mapping η defined by η(x) = xφ(x), x ∈ G, is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL(2, q), q ≡ 1 mod 4 and PGL(2, q), q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n, q), n > 2, and the linear groups GL(n, q), n ≥ 2, admit a complete mapping.
DALLA VOLTA, F., Gavioli, N. (1997). On the admissibility of some linear and projective groups in odd characteristic. GEOMETRIAE DEDICATA, 66(3), 245-254 [10.1023/A:1004924402272].
On the admissibility of some linear and projective groups in odd characteristic
DALLA VOLTA, FRANCESCA;
1997
Abstract
A bijective mapping φ: G → G defined on a finite group G is complete if the mapping η defined by η(x) = xφ(x), x ∈ G, is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL(2, q), q ≡ 1 mod 4 and PGL(2, q), q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n, q), n > 2, and the linear groups GL(n, q), n ≥ 2, admit a complete mapping.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
