Frucht's theorem

The set of fruit ( by Roberto fruit) is a set of the mathematical subfield of graph theory. He says that up to isomorphism, each group acting as automorphisms of a graph.

An automorphism of an undirected graph, where the vertex set and the edge set is a bijection with the property that two nodes are connected by an edge if and only if and are connected by an edge. The set of all automorphisms of is apparently a group and is called the automorphism group of.

Apparently for a edgeless graph or a complete graph is equal to the symmetric group of of. For all other graphs is a proper subgroup of. In extreme cases, such graphs are called asymmetric. The smallest number of nodes of an asymmetric graph is 6

Since by the theorem of Cayley each group is isomorphic to a subgroup of a symmetric group, the question arises whether each group acts as automorphisms of a graph. This question is answered positively by the set of fruit:

  • Set of fruit: there is a graph whose automorphism group is isomorphic to the group to each group.

This sentence has been stated and proved in 1938 by Roberto fruit for finite groups. The case of infinite groups was proved independently by J. de Groot (1959) and G. Sabidussi ( 1960).

710323
de