On the Norm of the Abelian p -Group-Residuals

Baojun Li, Yu Han, Lü Gong and Tong Jiang
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Baojun Li: School of Sciences, Nantong University, Nantong 226019, China
Yu Han: School of Sciences, Nantong University, Nantong 226019, China
Lü Gong: School of Sciences, Nantong University, Nantong 226019, China
Tong Jiang: School of Sciences, Nantong University, Nantong 226019, China

Mathematics, 2021, vol. 9, issue 8, 1-6

Abstract: Let G be a group. D p ( G ) = ? H ? G N G ( H ? ( p ) ) is defined and, the properties of D p ( G ) are investigated. It is proved that D p ( G ) = P [ A ] , where P = D ( P ) is the Sylow p -subgroup and A = N ( A ) is a Hall p ? -subgroup of D p ( G ) , respectively. Furthermore, it is proved in a group G that (1) D p ( G ) = 1 if and only if C G ( G ? ( p ) ) = 1 ; (2) O p ? ( D p ( G ) ) ? Z ? ( O p ( G ) ) and (3) if Z ( G ? ( p ) ) = 1 , then C G ( G ? ( p ) ) = D p ( G ) .

Keywords: finite group; abelian p-group residual; soluble group; normalizer (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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