On minimal non-p-closed groups and related properties
- Autores: Luise-Charlotte Kappe, Marcin Mazur, Gabriela Mendoza, Michael B. Ward
- Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 78, Fasc. 1, 2011, pág. 15
- Idioma: inglés
- DOI: 10.5486/pmd.2011.4779
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Resumen
Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup.
A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are p-closed. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups and homomorphic images are p-exponent closed. In this paper we characterize nite minimal non-p-closed groups and investigate the relationship between them and minimal non-p-exponent closed groups. In particular, we show that every minimal non-p-closed group is non- p-exponent closed and that minimal non-p-closed groups and simple minimal non-p- exponent closed groups have cyclic Sylow p-subgroups. Furthermore, given a prime p, we describe non-p-exponent closed groups of smallest order and we show that they coincide with non-p-closed groups of smallest order.
