Finite Groups in which every Subgroup of Order Divisible by p is Normal

The article introduces two generalizations of Dedekind groups known as pdDdkgroups and pdqHm-groups. In pdDdk-groups, subgroups with sizes that are multiples of a given prime p are normal, while in pdqHm-groups, these subgroups are permutable. It is initially established that these groups belong to either p0 -groups or supersolvable groups. It then proceeds to classify pdDdk-groups and pdqHmgroups. The article concludes with a discussion of minimal non-pdDdk-groups.