1Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México. Email: lhuerta@fismat.umich.mx
2Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México. Email: ariel@zarzamora.com.mx
3Universidad Nacional Autónoma de México, Morelia, México. Email: graggi@matmor.unam.mx
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Let G and Q be groups with isomorphic tables of marks, and for each subgroup H of G, let H' denote a subgroup of Q assigned to H under an isomorphism between the tables of marks of G and Q. We prove that if H is cyclic/elementary abelian/maximal/the Frattini subgroup/the commutator subgroup, then H' has the same property. However, we give examples where H is abelian and H' is not, and where H is the centre of G and H' is not the centre of Q. For this we construct (using GAP) the smallest example of non-isomorphic groups with isomorphic tables of marks.
Key words: Group representation, Burnside rings, table of marks.
Sean G y Q grupos con tablas de marcas isomorfas, y para cada subgrupo H de G, sea H' un subgrupo de Q asignado a H bajo un isomorfismo entre las tablas de marcas de G y Q. Demostramos que si H es cíclico/elemental abeliano/maximal/el subgrupo de Frattini/el subgrupo conmutador, entonces H' tiene la misma propiedad. Sin embargo, damos ejemplos donde H es abeliano y H' no lo es y donde H es el centro de G y H' no es el centro de Q. Para esto construimos (usando GAP) el menor ejemplo de grupos no isomorfos con tablas de marcas isomorfas.
Palabras clave: Representación de grupos, anillo de Burnside, tabla de marcas.
References
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[6] F. Luca and A. Raggi-Cárdenas, `Composition factors from the table of marks´, Journal of Algebra 244, (2001), 737-743. [ Links ]
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[8] A. Raggi-Cárdenas and L. Valero-Elizondo, `Two non-isomorphic groups of order 96 with isomorphic tables of marks and non-corresponding centres and abelian subgroups´, Communications in Algebra 37, (2009), 209-212. ISSN: 0092-7872, DOI: 10.1080/00927870802243614 [ Links ]
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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCMv43n2a05,
AUTHOR = {Huerta-Aparicio, Luis and Molina-Rueda, Ariel and Raggi-Cárdenas, Alberto and Valero-Elizondo, Luis},
TITLE = {{On some invariants preserved by isomorphisms of tables of marks}},
JOURNAL = {Revista Colombiana de Matemáticas},
YEAR = {2009},
volume = {43},
number = {2},
pages = {165-174} ]]>
}