We associate a graph Gamma(G) to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G Cyc (G) as vertex set, where Cyc (G ) = {x epsilon vertical bar (x , y ) is cyclic for all y epsilon G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of (G is. nite if and only if Gamma(G) has no in. nite clique. We prove that if G is a. nite nilpotent group and H is a group with Gamma(G) congruent to Gamma(H) H and vertical bar Cyc (G) vertical bar = vertical bar Cyc(H)vertical bar = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are " unique", i. e., if Gamma(G) congruent to Gamma(H) for some group H, then G congruent to H. In view of these examples, we conjecture that every. nite nonabelian simple group has a unique noncyclic graph. Also we give some examples of. nite noncyclic groups G with the property that if (G congruent to Gamma(H) for some group H, then vertical bar G vertical bar = vertical bar H vertical bar. These suggest the question whether the latter property holds for all. nite noncyclic groups.
展开▼
