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Unformatted text preview: Norms and Distances over Finite Groups Vladimir Batagelj * University of Ljubljana, Department of Mathematics Jadranska 19, 61 111 Ljubljana, Yugoslavia Current version: October 21, 1990 Abstract In the paper norms and distances over finite groups are studied. A characteri zation of ultrametric interval group norms is given. It is proved that the maximal range of interval group norms is attained on groups which are the powers of the cyclic group of order two. Key words : finite groups, interval group norms, distances, ordered partitions, Hamming norm, (generalized) Lee norm, SharmaKaushik partitions. Math. Subj. Class. (1985) : 20 D 60, 54 E 35, 11 T 71, 94 B 60 During professor Sharma’s presentation of his paper Association and other Schemes Related to SharmaKaushik Class of Distances over Finite Rings [2] we noticed that in the construction of a distance, using the weight induced by a partition, we need only the structure of the group ( ZZ q , + q ). The idea of the construction can be extended to any (finite) group provided that the weight is a ’group norm’. We were convinced that group norms would already have been studied; but it took us several days to finally find a reference on this subject [8]. Another reference [3] was pointed out by one of the referees. The proposition 5 was the only result that we did not know before reading [8]. Although some basic facts about group norms can also be found in [8], we decided to reproduce them to make our paper easier to read. 1 Group norms Let ( A, • , e ) be a finite group with a neutral element e . We denote the inverse element of a ∈ A by a . The mapping w : A → IR is a (group) norm if it has the following properties: * Supported in part by the Research Council of Slovenia, Yugoslavia; and by the Yugoslav Federal Grant P339. 1 E. ∀ a ∈ A : ( w ( a ) = 0 ⇔ a = e ) S. ∀ a ∈ A : w ( a ) = w ( a ) and N. ∀ a, b ∈ A : ( w ( a ) + w ( b ) ≥ w ( a • b )) If the mapping w has the properties E, S and U. ∀ a, b ∈ A : (max( w ( a ) , w ( b )) ≥ w ( a • b )) we shall call it an ultrametric (group) norm [4, p. 39]. We shall denote the normed group by ( A, • , e ; w ). Several proofs for ultrametric norms are essentially the same as for ordinary norms. In these cases we shall use the symbol ⊕ to denote the operation max or addition. Let us list some properties of group norms: Proposition 1 For all a ∈ A : w ( a ) ≥ . Proposition 2 Every ultrametric group norm is also a group norm. From group theory we know that the product ( A × B, ?, e ) of two groups ( A, • , e A ) and ( B, ◦ , e B ), where ( a, b ) ? ( α, β ) = ( a • α, b ◦ β ) is also a group. The following proposition shows that we can extend norms from groups to their products....
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