A new conjecture concerning admissibility of groups
A finite group is said to be admissible if it has a permutation mapping \(g\to \theta (g)\) such that the mapping \(g\to g\cdot \theta (g)\) is also a permutation. A necessary condition for a finite group to be admissible is that the product of all its elements in some order is equal to the identity. Another necessary condition is that its Sylow 2-subgroups are not cyclic; both conditions are sufficient for soluble groups. Since the second condition is satisfied for all non-soluble groups, the authors propose as new conjecture: All non-soluble groups are admissible. And the authors prove: Theorem: If G is a non-soluble group there exists an ordering of the elements of G the product of which is the identity element of the group (so, for non-soluble group both necessary conditions are satisfied).
scientific article; zbMATH DE number 4125677