The Functional Equation max{ ? ( xy ), ? ( xy -1 )}= ? ( x ) ? ( y ) on Groups and Related Results

Muhammad Sarfraz, Qi Liu and Yongjin Li
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Muhammad Sarfraz: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
Qi Liu: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
Yongjin Li: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Mathematics, 2021, vol. 9, issue 4, 1-10

Abstract: This research paper focuses on the investigation of the solutions ? : G ? R of the maximum functional equation max { ? ( x y ) , ? ( x y ? 1 ) } = ? ( x ) ? ( y ) , for every x , y ? G , where G is any group. We determine that if a group G is divisible by two and three, then every non-zero solution is necessarily strictly positive; by the work of Toborg, we can then conclude that the solutions are exactly the e | ? | for an additive function ? : G ? R . Moreover, our investigation yields reliable solutions to a functional equation on any group G , instead of being divisible by two and three. We also prove the existence of normal subgroups Z ? and N ? of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G / N ? .

Keywords: additive function; normal subgroup; strictly positive solution; commutators; maximum functional equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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