|
UPSI Digital Repository (UDRep)
|
|
| ||||||||||||||||||||||
| Abstract : Universiti Pendidikan Sultan Idris |
| 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. |
| This material may be protected under Copyright Act which governs the making of photocopies or reproductions of copyrighted materials. You may use the digitized material for private study, scholarship, or research. |