The Neighbourhood of Dihedral 2-groups

We examine two particular constructions that derive from a 2-group G = G(·) another 2-group G(∗) for the case when G(·) is one of D2n , SD2n , Q2n . The constructions (the cyclic one and the dihedral one) have the property that x ∗ y = x · y for exactly 3/4 of all pairs (x, y) ∈ G × G. If G(◦) and G(∗) are such 2-groups that x ◦ y 6= x ∗ y for less then a quarter of all pairs (x, y) ∈ G × G, then G(◦) and G(∗) are isomorphic [4]. This makes of a special interest situations when G(◦) and G(∗) are not isomorphic and x ◦ y 6= 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(◦) ∼= G1 and G(∗) ∼= G2 of the above property. It seems to be a difficult task to decide for which G1 and G2 such G(◦) and G(∗) exist. Nevertheless, there exist general constructions that allow 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 [5] 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 presented already 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(∗). 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 7→ 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 integer i ∈ Z. Consider a group G = G(·) with a normal subgroup S, G/S ∼= C2m , and with an element h ∈ Z(G) ∩ S. For every generating coset α ∈ G/S define the group G(∗) = G[α, h] by x ∗ y = xyh, where x ∈ α, y ∈ α and i, j ∈ M . See [5] or [3] for a proof that G(∗) is really a group. We have described the cyclic construction. The dihedral construction follows; its justification can be found in [5] and [3] again. Work supported by by Grant Agency of Charles University, grant number 269/2001/BMAT/MFF. The second and the third authors were also supported by the institutional grants MSM 113200007 and MSM 210000010, respectively.