Maximal irredundant coverings of some finite groups

The aim of this research is to contribute further results on the coverings of some finite groups. Only non-cyclic groups are considered in the study of group coverings. Since no group can be covered by only two of its proper subgroups, a covering should consist of at least 3 of its proper subgroups. If a covering contains n (proper) sub- groups, then the set of these subgroups is called an n-covering. The covering of a group G is called minimal if it consists of the least number of proper subgroups among all coverings for the group; i.e. if the minimal covering consists of m proper subgroups then the notation used is σ (G) = m. A covering of a group is called irredundant if no proper subset of the covering also covers the group. Obviously, every minimal cover- ing is irredundant but the converse is not true in general. If the members of the covering are all maximal normal subgroups of a group G, then the covering is called a maximal covering. Let D be the intersection of all members in the covering. Then the covering is said to have core-free intersection if the core of D is the trivial subgroup. A maxi- mal irredundant n-covering with core-free intersection is known as a Cn-covering and a group with this type of covering is known as a Cn-group. This study focuses only on the minimal covering of the symmetric group S9 and the dihedral group Dn for odd n ≥ 3; on the characterization of p-groups having a Cn-covering for n ∈ {10, 11, 12}; and the characterization of nilpotent groups having a Cn-covering for n ∈ {9, 10, 11, 12}. In this thesis, a lower bound and an upper bound for σ (S9) is established. (However, later it was found that the exact value for σ (S9) = 256 has already been discovered in 2016.) For the dihedral groups Dn where n is odd and n ≥ 3, results were presented in two classifications, i.e. the prime n and the odd composite n. For the p-groups, it was found that the only p-groups with Cn-coverings for n ∈ {10, 11, 12} are those isomor- phic to some elementary abelian groups of certain orders and the results established the concrete proof of the groups. It was also found that some p-groups have all three pos- sible types of coverings and some others have two of the three types of coverings. For the nilpotent groups, it was found that for n ∈ {10, 11, 12}, the nilpotent groups hav- ing Cn-coverings are exactly the p-groups obtained earlier; no other nilpotent groups were found to have Cn-coverings for n ∈ {10, 11, 12}. The nilpotent groups having a C9-covering are also isomorphic to some elementary abelian groups of certain orders.