6533b853fe1ef96bd12acd5b
RESEARCH PRODUCT
Verbal sets and cyclic coverings
Giovanni CutoloChiara Nicoterasubject
Discrete mathematicsCommutatorgroup wordAlgebra and Number TheorySubgroup coveringscommutatorComputer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing)Central seriescoveringSet (abstract data type)Verbal subgroupsVerbal subgroupCharacteristic subgroupGroup theoryLower central seriesFinite setWord (group theory)Group theoryCyclic subgroupsMathematicsdescription
Abstract We consider groups G such that the set of all values of a fixed word w in G is covered by a finite set of cyclic subgroups. Fernandez-Alcober and Shumyatsky studied such groups in the case when w is the word [ x 1 , x 2 ] , and proved that in this case the corresponding verbal subgroup G ′ is either cyclic or finite. Answering a question asked by them, we show that this is far from being the general rule. However, we prove a weaker form of their result in the case when w is either a lower commutator word or a non-commutator word, showing that in the given hypothesis the verbal subgroup w ( G ) must be finite-by-cyclic. Even this weaker conclusion is not universally valid: it fails for verbose words.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2010-10-01 | Journal of Algebra |