English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The non-abelian groups which contain only abelian subgroups have been determined and many of their fundamental properties have been noted, f The present paper is devoted to a determination and a study of the non-abelian groups which involve only abelian and hamiltonian subgroups. It will be found that there are only two exceptions to the following theorem : If a group contains at least one hamiltonian subgroup and if all its subgroups are either abelian or hamiltonian the group itself is hamiltonian. The two exceptional groups are of orders 16 and 24 respectively, and are completely determined in the following paragraphs. Let G represent any group in which every subgroup is either abelian or hamiltonian and suppose that it is represented as a transitive substitution group of the lowest possible degree. If it is imprimitive its systems of imprimitivity are permuted according to some primitive group. To prove that G is solvable it is only necessary to prove that this primitive group is composite since it may be simply isomorphic with G and since every subgroup of G is solvable. The primitive group in question is of class n — 1, n being its degree, since two of its maximal subgroups of degree n—1 can have only the identity in common. \ As every group of class n — 1 and degree n is composite G is solvable. In the preceding paragraph it is not assumed that G necessarily contains a hamiltonian subgroup. This assumption may, however, be made since the nonabelian groups in which every subgroup is abelian are known. Hence we shall assume in what follows that G contains at least one hamiltonian subgroup. Its order can therefore not be the power of an odd prime number. We shall consider all the possible G'a under two headings as their orders are a power of 2 or involve more than one prime factor.

The groups in which every subgroup is either abelian or Hamiltonian