On the \(R\)-sequenceability of dicyclic groups
A finite group \(G\) of order \(n\) is said to be \(R\)-sequenceable if its elements can be ordered in a sequence, \(a_ 1,a_ 2,\dots,a_ n\), such that all of the partial products \(b_ 1 = a_ 1\), \(b_ 2 = a_ 1a_ 2, \dots, b_{n-1} = a_ 1a_ 2 \dots a_{n-1}\) are different and so that the product \(b_ n = a_ 1a_ 2 \dots a_ n = e\), where \(e\) denotes the identity element of \(G\). The dicyclic group \(Q_{2n}\) of order \(4n\) is defined by \(Q_{2n} = gp\{a,b: a^{2n} = e,\;b^ 2 = a^ n,\;ab = ba^{-1}\}\). In this paper, the author investigates the \(R\)-sequenceability of \(Q_{2n}\). The following results are proved: (1) \(Q_{2n}\) is \(R\)-sequenceable, if \(n \equiv 0 \bmod 4\). (2) \(Q_{2n}\) is \(R\)-sequenceable, if \(n + 1\) is a prime which has \(-2\) as a primitive root. In order to prove the above two results, the author gives a sufficient condition for the \(R\)-sequenceability of the dicyclic group \(Q_{2n}\) and gives two specific constructions by which the condition can be met when (1) \(n = 4k\) or (2) \(n + 1\) is a prime \(p\) say and \(-2\) is a primitive root modulo \(p\). The author conjectures that the condition can be met for all even integers \(n > 2\).
scientific article; zbMATH DE number 562684