Maximal Irredundant Coverings Of Some Finite Groups - Tarmizi, Rawdah Adawiyah (2018)
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) subgroups, 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 s(G) = m. A covering of a group is called irredundant if no proper subset of the covering also covers the group. Obviously, every minimal covering 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 maximal 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 2 f10;11;12g; and the characterization of nilpotent groups having a Cn-covering for n 2 f9;10;11;12g. In this thesis, a lower bound and an upper bound for s(S9) is established.