- Volume:7 Issue:2, 2018
- تاریخ انتشار: 1396/11/09
- تعداد عناوین: 6
Pages 1-8Let G be a finite group and let P=P1, ,Pm be a sequenceý ýof Sylow pi-subgroups of Gý, ýwhere p1, ,pm are the distinctý ýprime divisors of |G|ý. ýThe Sylow multiplicity of g∈ýýG in P is the number of distinct factorizations g=g1⋯ýýgm such that gi∈Piý. ýWe review properties of the solvableý ýradical and the solvable residual of G which are formulated in terms ofý ýSylow multiplicitiesý, ýand discuss some related open questionsý.Keywords: ?Sylow sequences?, ?Sylow multiplicities?, ?Solvable radical?, ?Solvable residual
Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding propertiesPages 9-16In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation iný ýcertain universes of groups with some finiteness propertiesý.Keywords: ?group?, ?subgroup? ?embedding property?, ?T-group?, ?FC?-group?, ?group without infinite? ?simple sections
Pages 17-24We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.Keywords: finite groups, Conjugacy classes, Factorised groups
Pages 25-29ýThe aim of this survey article is to present some structural results about of groups whose Sylow p-subgroups are metacylic (p a prime)ý. ýA complete characterisation of non-nilpotent groups whose 2-generator subgroups are metacyclic is also presentedý.Keywords: Finite group, Sylow subgroups, metacyclic groups, 2-generator groups
Pages 31-44An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix A of CG there exists a matrix U (independent of A) such that U−1AU=diag(T1,T2, ,Tr) for block matrices Ti of fixed size si×si where r is the number of conjugacy classes of G and si are the ranks of the group ring matrices of the primitive idempotents.
Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping A↦P−1AP (fixed P) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of G may be derived using the blocks in the images of the group elements.
For a finite abelian group Q an explicit matrix P is given which diagonalises any group ring matrix of CQ. The characters of Q and the character table of Q may be read off directly from the rows of the diagonalising matrix P. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.Keywords: group, ring, Representation
Pages 45-50Let L be a free Lie algebra of rank r≥2 over a field F and let Ln denote the degree n homogeneous component of Lý. ýBy using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field Fý, ýwe determine the dimension of [L2,L2,L1]ý. ýMoreoverý, ýby this methodý, ýwe show that the dimension of [L2,L2,L1] over a field of characteristic 2 is different from the dimension over a field of characteristic other than 2.Keywords: Free Lie algebra, homogeneous, fine homogeneous components, free centre-by-metabelian Lie algebra, second derived ideal