The power graph of a torsion-free group
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The power graph of a torsion-free group Peter J. Cameron1
· Horacio Guerra1,2 · Šimon Jurina1
Received: 1 July 2017 / Accepted: 19 February 2018 / Published online: 28 February 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The power graph P(G) of a group G is the graph whose vertex set is G, with x and y joined if one is a power of the other; the directed power graph − → P (G) has the same vertex set, with an arc from x to y if y is a power of x. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, we consider power graphs of torsion-free groups. Our main results are that, for torsion-free nilpotent groups of class at most 2, and for groups in which every non-identity element lies in a unique maximal cyclic subgroup, the power graph determines the directed power graph up to isomorphism. For specific groups such as Z and Q, we obtain more precise results. Any isomorphism P(Z) → P(G) preserves orientation, so induces an isomorphism between directed power graphs; in the case of Q, the orientations are either all preserved or all reversed. We also obtain results about groups in which every element is contained in a unique maximal cyclic subgroup (this class includes the free and free abelian groups), and about subgroups of the additive group of Q and about Qn . Keywords Power graph · Directed power graph · Torsion-free group
Horacio Guerra and Šimon Jurina acknowledge funding from the School of Mathematics and Statistics for summer internships during which this research was carried out.
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Peter J. Cameron [email protected]
1
University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK
2
Present Address: School of Mathematics and Statistics, Newcastle upon Tyne NE1 7RU, UK
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84
J Algebr Comb (2019) 49:83–98
Mathematics Subject Classification 05C25 · 20F99
1 Introduction Let G be a group. Then the power graph of G is the graph P(G) with vertex set V (P(G)) = G and edge set E(P(G)) = {{x, y} : (∃ n ∈ Z \ {0}) (x = y n or y = x n )}. − → The directed power graph of G is the directed graph P (G) with vertex set − → V ( P (G)) = G and arc set − → E( P (G)) = {(x, y) : (∃ n ∈ Z \ {0}) (y = x n )}. − → Thus, P (G) is an orientation of P(G). When x and y are connected in P(G), we write x ∼ y. If a is a power of b in G, we denote this by b → a. In a graph , we denote the set of neighbours of a vertex x by N (x); in a directed graph, we denote the set of in-neighbours of x by I (x), and the set of out-neighbours by O(x). Since we will always be considering power graphs of groups, we denote N G (x) for the set of neighbours of x in P(G), and IG (x), OG (x) for the sets of in− → and out-neighbours of x in P (G). The directed power graph was first defined in the context of semigroups by K
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