The Neighbourhood of Dihedral 2-Groups
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The Neighbourhood of Dihedral 2-Groups MARTIN BÁLEK, ALEŠ DRÁPAL and NATALIA ZHUKAVETS Dept. of Algebra, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic. e-mail: [email protected], {drapal,natalia}@karlin.mff.cuni.cz Abstract. We examine two particular constructions that derive from a 2-group G = G(·) another 2group G(∗) for the case when G(·) is one of D2n , SD2n , Q2n . The constructions (cyclic and dihedral) have the property that x ∗ y = x · y for exactly 3/4 of all pairs (x, y) ∈ G × G. Mathematics Subject Classifications (2000): primary: 20D60, secondary: 05B15. Key words: 2-group, Hamming distance, group multiplication table.
If G(◦) and G(∗) are such 2-groups that x ◦ y = x ∗ y for less then a quarter of all pairs (x, y) ∈ G × G, then G(◦) and G(∗) are isomorphic [3]. This makes of special interest situations when G(◦) and G(∗) are not isomorphic and x ◦y = x ∗y for exactly one-quarter of all pairs. Say that groups G1 and G2 can be placed in quarter distance if there exist G(◦) ∼ = G2 of the above property. It seems to be a difficult task = G1 and G(∗) ∼ to decide for which G1 and G2 such G(◦) and G(∗) exist. Nevertheless, there exist general constructions that allow us to derive from a group G = G(·) another group G(∗) such that x · y = x ∗ y for exactly three-quarters of all pairs (x, y) ∈ G × G. The constructions we shall examine here are called cyclic and dihedral. They are described in [4] and it seems that all presently known pairs of 2-groups that can be placed in quarter distance can be interpreted in terms of these constructions. It was observed in [1] that from a dihedral group D2n+1 (which is of nilpotency class n) one can obtain, by one of these constructions, a group of nilpotency class n − m, for every m n/2. This can be regarded as quite surprising, since it shows that two 2-groups can share a large part of their multiplication tables (i.e., three-quarters) and still possess rather distinct nilpotency degrees. This paper describes all groups that can be obtained by the mentioned constructions from D2n+1 and Q2n+1 , n 2, and from SD2n+1 , n 3. It generalizes [7], where only the simplest case of the constructions was considered. Some examples of 2-groups in quarter distance involving the mentioned groups were already presented in [2]. Our approach is systematic and covers all the known cases. We write x ∗ to denote the inverse of x in a group G(∗). Work supported by by Grant Agency of Charles University, grant number 269/2001/B-
MAT/MFF. The second and the third authors were also supported by the institutional grants MSM 113200007 and MSM 210000010, respectively.
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´ MARTIN BALEK ET AL.
1. Constructions and Isomorphisms Denote by Ck the cyclic group of order k, and by D2k the dihedral group of order 2k. If k = 2, then D2k means C2 × C2 . Fix m 1 and put M = {−2m−1 + 1, . . . , 0, 1, . . . , 2m−1 }. Denote by µ the permutation i → 1 − i of M, and define σ by σ (i) = 0 if i ∈ M, σ (i) = 1 if i > 2m−1 and σ (i) = −1 if i −2m−1 , for every
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