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

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
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Academic Editor: Yeol Je Cho
Mathematics 2021, 9(4), 382; https://doi.org/10.3390/math9040382
Received: 19 December 2020 / Revised: 8 February 2021 / Accepted: 9 February 2021 / Published: 14 February 2021
This research paper focuses on the investigation of the solutions χ:GR of the maximum functional equation max{χ(xy),χ(xy1)}=χ(x)χ(y), for every x,yG, 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|α| for an additive function α:GR. 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 Zχ and Nχ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G/Nχ. View Full-Text
Keywords: additive function; normal subgroup; strictly positive solution; commutators; maximum functional equation additive function; normal subgroup; strictly positive solution; commutators; maximum functional equation
MDPI and ACS Style

Sarfraz, M.; Liu, Q.; Li, Y. The Functional Equation max{χ(xy),χ(xy-1)}=χ(x)χ(y) on Groups and Related Results. Mathematics 2021, 9, 382. https://doi.org/10.3390/math9040382

AMA Style

Sarfraz M, Liu Q, Li Y. The Functional Equation max{χ(xy),χ(xy-1)}=χ(x)χ(y) on Groups and Related Results. Mathematics. 2021; 9(4):382. https://doi.org/10.3390/math9040382

Chicago/Turabian Style

Sarfraz, Muhammad; Liu, Qi; Li, Yongjin. 2021. "The Functional Equation max{χ(xy),χ(xy-1)}=χ(x)χ(y) on Groups and Related Results" Mathematics 9, no. 4: 382. https://doi.org/10.3390/math9040382

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