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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.56 no.2 Bogotá July/Dec. 2022 Epub Feb 06, 2024
https://doi.org/10.15446/recolma.v56n2.108379
Original articles
On the Fischer matrices of a group of shape 2 1+2n +:G
Sobre las matrices de Fischer de un grupo de la forma 2 1+2n +:G
1 Nelson Mandela University, Gqeberha, South Africa
In this paper, the Fischer matrices of the maximal subgroup G = 21+8 +: (U 4(2):2) of U 6(2):2 will be derived from the Fischer matrices of the quotient group Q = G/Z(21+8 +) ( 28: (U 4(2):2), where Z(21+8 +) denotes the center of the extra-special 2-group 21+8 +. Using this approach, the Fischer matrices and associated ordinary character table of G are computed in an elegantly simple manner. This approach can be used to compute the ordinary character table of any split extension group of the form 2 1+2n +: G, n ∈ N, provided the ordinary irreducible characters of 2 1+2n + extend to ordinary irreducible characters of its inertia subgroups in 2 1+2n +:G and also that the Fischer matrices M(g i ) of the quotient group 2 1+2n +: G/Z(2 1+2n +) ( 2 2n : G are known for each class representative g i in G.
Keywords: split extension; extra-special p-group; irreducible projective characters; Schur multiplier; inertia factor groups; Fischer matrices
En este artículo, las matrices de Fischer del subgrupo maximal G = 21+8 +: (U 4(2):2) de U 6(2):2 serán derivadas a partir de las matrices de Fischer del grupo cociente Q = G/Z(21+8 +) ( 28: (U 4(2):2), donde Z(21+8 +) denota el centro del grupo 2-extra especial 21+8 +. Usando este enfoque, las matrices de Fischer y la tabla de caracteres asociadas de G son calculados de una manera elegante y simple. Este enfoque se puede utilizar para calcular la tabla de caracteres de cualquier extensión escindida de la forma 2 1+2n +:G, n ∈ N, siempre y cuando los caracteres irreducibles ordinarios de 2 1+2n + se extiendan a caracteres irreducibles ordinarios de sus subgrupos de inercia en 2 1+2n +:G y también que las matrices de Fischer M(g i ) del grupo cociente 2 1+2n +: G/Z(2 1+2n +) ( 2 2n : G sean conocidas para cada representante de clase g i en G.
Palabras clave: extensión escindida; p-grupo extra especial; caracteres proyectivos irreducibles; multiplicador de Schur; inertia factor groups; matrices de Fischer
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Received: October 17, 2021; Accepted: January 06, 2023
This is an open-access article distributed under the terms of the Creative Commons Attribution License
