CLT-groups with cyclic or abelian subgroups

TL;DR

This paper classifies numbers for which all groups of that order are cyclic or abelian CLT groups, explores their properties, and introduces a new function to measure the CLT-degree of non-cyclic groups.

Contribution

It provides a complete classification of CCLT and ACLT numbers, analyzes properties of cyclic and abelian CLT groups, and introduces the CCLT-degree function.

Findings

Classified all CCLT and ACLT numbers.

Showed CCLT and ACLT groups are supersolvable.

Introduced and studied the CCLT-degree function.

Abstract

A finite group is called a CLT-group if it contains a subgroup corresponding to every divisor of the order of the group. It is said to be a Cyclic (Abelian) CLT group if it contains a cyclic (abelian) subgroup corresponding to every proper divisor of the order of the group. A natural number is said to be a CCLT (ACLT) number if every group of that order is a cyclic (abelian) CLT group. In this work, we classify all CCLT and ACLT numbers and study various properties of Cyclic (Abelian) CLT groups. We also show that the classes of CCLT and ACLT groups are contained in the class of supersolvable groups. Moreover, we introduce the function CCLT-degree on the set of non-cyclic finite groups and study the properties of this function.