This paper investigates the approach of presenting groups by generators and relations from an original angle. It starts by interpreting this familiar concept with the novel notion of “formal words” created by juxtap...This paper investigates the approach of presenting groups by generators and relations from an original angle. It starts by interpreting this familiar concept with the novel notion of “formal words” created by juxtaposing letters in a set. Taking that as basis, several fundamental results related to free groups, such as Dyck’s Theorem, are proven. Then, the paper highlights three creative applications of the concept in classifying finite groups of a fixed order, representing all dihedral groups geometrically, and analyzing knots topologically. All three applications are of considerable significance in their respective topic areas and serve to illustrate the advantages and certain limitations of the approach flexibly and comprehensively.展开更多
In this note, all groups are finite and the notations and termentary are standard, inaddition, given a finite G group. let π(G) and π(|G|) denote the set of primes dividing |G|.
基金supported by NSFC(No.11271301,No.11471055,No.11661031)Scientific Research Foundation of Sichuan Provincial Education Department(No.14ZA0314,No.15ZA0181)the Scientific Research Foundation of CUIT(No.J201418)
文摘This paper investigates the approach of presenting groups by generators and relations from an original angle. It starts by interpreting this familiar concept with the novel notion of “formal words” created by juxtaposing letters in a set. Taking that as basis, several fundamental results related to free groups, such as Dyck’s Theorem, are proven. Then, the paper highlights three creative applications of the concept in classifying finite groups of a fixed order, representing all dihedral groups geometrically, and analyzing knots topologically. All three applications are of considerable significance in their respective topic areas and serve to illustrate the advantages and certain limitations of the approach flexibly and comprehensively.
文摘In this note, all groups are finite and the notations and termentary are standard, inaddition, given a finite G group. let π(G) and π(|G|) denote the set of primes dividing |G|.
