S-PERMUTABILITY, SEMIPERMUTABILITY, c-NORMALITY AND c-PERMUTABILITY OF SUBGROUPS IN FINITE GROUPS
| dc.contributor.author | Al-Sharo, Doa a Mustafa | |
| dc.date.accessioned | 2015-07-30T00:20:40Z | |
| dc.date.available | 2015-07-30T00:20:40Z | |
| dc.date.issued | 2014-08 | |
| dc.description.abstract | All groups and subgroups considered in this thesis are nite. In this thesis, some new theories and related proofs on S-permutable (also referred to as Sylow per- mutable), semipermutable, c-normal and c-permutable subgroups of a nite group G are discussed. More than one topic are focused here, some of which are, the relationship between classes of S-permutable embedded subgroups and PST-groups, the semiper- mutable subgroups, and the condition when c-normal and c-permutable subgroups are transitive. New ndings reveal that PST-groups, the class of groups in which S- permutability is a transitive relation, coincides with the class of nite solvable groups in which every p-subgroup is S-permutably embedded. Some facts were developed to prove this result, in particular, a simpli ed proof is given to prove that if a subgroup is S-permutably embedded in the group then it should be S-permutably embedded in every subgroup that contains it. A dual result is also established for a group called the PSTc-group where "c" denotes the special condition on the nilpotent residual. Research on semipermutability of subgroups and their transitivity relationships have been extensively explored in previous years. When semipermutability in a group G is a transitive relation, G is called a BT-group. When all subnormal subgroups of G are semipermutable, G is called an SP-group. It has been shown that all BT-groups are SP-groups. However, an SP-group is not necessarily a BT-group as shown in the the- sis. For the condition in which c-normality is a transitive relation, a new class of group called CT-groups is introduced and some related properties and theorems to this group are proved. Some new results related to CPT-groups, which are groups in which c- permutability is a transitive relation, are also given. This thesis also contains the proof that there exists a c-permutable subgroup which is not c-normal. New characteriza- tions of the following groups and subgroups are de ned and proved in this thesis which are: the nite solvable PST-groups, S-permutably embedded subgroups, BT-groups, semipermutable subgroups, seminormal subgroups, nite solvable CT-groups, mini- mal non-CT-groups, minimal non-CPT-groups and groups with all its subnormal sub- groups c-normal (called SCN groups), and groups with all its subnormal subgroups c- permutable (called SCp-groups). | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/869 | |
| dc.subject | S-PERMUTABILITY | en_US |
| dc.subject | SEMIPERMUTABILITY | en_US |
| dc.subject | c-NORMALITY | en_US |
| dc.subject | c-PERMUTABILITY | en_US |
| dc.title | S-PERMUTABILITY, SEMIPERMUTABILITY, c-NORMALITY AND c-PERMUTABILITY OF SUBGROUPS IN FINITE GROUPS | en_US |
| dc.type | Thesis | en_US |
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