Some questions about subgroups and structures of finite groups
W. Guo
In this talk, we give the answers to the following questions:
Question 1. What is the structure of the finite groups \(G\) in which
every subgroup can be written as an intersection of subgroups of
prime power indexes?
Question 2. What is the structure of the finite groups \(G\) in which
every subnormal subgroup can be written as an intersection of
subnormal subgroups of prime power indexes?
Question 3. Is the intersection of all maximal \({\cal F}\)-subgroups of a finite group \(G\) equal to the \({\cal F}\)-hypercentre of \(G\), for any hereditary saturated formation
\({\cal F}\) ?
Question 4. If the answer to Question 3 is negative, can one give a criterion for the intersection of
all maximal \({\cal F}\)-subgroups of a finite group \(G\) to be equal to
the \({\cal F}\)-hypercentre of \(G\), for any hereditary saturated formation
\({\cal F}\) ?
Question 5. What is the structure of the finite groups in which the maximal
subgroups, \(2\)-maximal subgroups, and \(3\)-maximal subgroups are pairwise permutable?
