Author
Liangcai Zhang
Bio: Liangcai Zhang is an academic researcher from Chongqing University. The author has contributed to research in topics: Order (group theory) & Nilpotent group. The author has an hindex of 1, co-authored 1 publications receiving 8 citations.
Topics: Order (group theory), Nilpotent group, Finite group, Nilpotent
Papers
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TL;DR: In this article, a framework for generalisations of Baer's norm has been given for a class of finite nilpotent groups, where the C -norm κ C (G ) of a finite group G is defined as the intersection of the normalisers of the subgroups of G not in C.
16 citations
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15 Nov 2016
TL;DR: In this article, the authors specify all the known findings related to the norms of the group and their generalizations, and special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups.
Abstract: In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups.
7 citations
TL;DR: In this paper, the influence on a group G of the behaviour of its metanorm M (G ), defined as the intersection of all normalizers of non-abelian subgroups of G, has been studied and it is shown that G is metahamiltonian unless the order of the commutator subgroup of M(G ) is the square of a prime number.
7 citations
TL;DR: In this paper, the authors studied the embedding properties of the metanorm of a group, defined as the intersection of all normalizers of non-abelian subgroups, and proved that the norm is always contained in the second term of the upper central series of the group.
Abstract: Abstract The norm of a group was introduced by R. Baer as the intersection of all normalizers of subgroups, and it was later proved that the norm is always contained in the second term of the upper central series of the group. The aim of this paper is to study embedding properties of the metanorm of a group, defined as the intersection of all normalizers of non-abelian subgroups. The metanorm is related to the so-called metahamiltonian groups, i.e. groups in which all non-abelian subgroups are normal, and it is known that every locally graded metahamiltonian group is finite over its second centre. Among other results, it is proved here that if G is a locally graded group whose metanorm M is not nilpotent, then M′/M′′{M^{\\prime}/M^{\\prime\\prime}} is a small eccentric chief factor and it is the only obstruction to a strong hypercentral embedding of M in G.
4 citations
TL;DR: In this article, generalized norms for different systems of infinite and noncyclic subgroups in nonperiodic groups are considered and the conditions under which these norms are satisfied are established.
Abstract: The authors consider generalized norms for different systems of infinite and noncyclic subgroups in nonperiodic groups. Relations between these norms are established. The conditions under which the...
4 citations
TL;DR: In this article, the authors classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal, which are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.
Abstract: We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.
3 citations