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Title: | Conjugacy classes of centralizers in groups |
Authors: | SINGH, ANUPAM KUMAR PARTHASARATHY, BHARGAVI Dept. of Mathematics 20141036 |
Keywords: | 2019 Mathematics |
Issue Date: | Apr-2019 |
Abstract: | Two elements in a group $G$ are said to be in the same $z$-class or $z$-equivalent if their centralizers are conjugate in $G$. This is an equivalence relation on $G$ and provides a partition of $G$ into disjoint equivalence classes. The structure of centralizers and their conjugacy classes provides important insight into the group structure. Although $z$-equivalence is a weaker relation than conjugacy, it is interesting to note that there are in finite groups which have infinitely many conjugacy classes but finitely many $z$-classes. In fact, the finiteness of $z$-classes in algebraic groups and Lie groups is an interesting problem. We have studied the structure of $z$-classes for symmetric groups $S_n$, general linear groups $GL_n(\mathbb{F})$ and general affine groups $GA_n(\mathbb{F})$ and have proven that there are finitely many $z$-classes, for $n \leq$ 5 in $S_n$ and when $\mathbb{F}$ has finitely many extensions, in the latter cases. We also investigate the idea that there is a relation between the finiteness of $z$-classes and and the intuitive understanding of the finiteness of "dynamical types" of transformations in geometry through group actions. |
URI: | http://dr.iiserpune.ac.in:8080/xmlui/handle/123456789/2961 |
Appears in Collections: | MS THESES |
Files in This Item:
File | Description | Size | Format | |
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Conjugacy classes of centralizers in groups-signed.pdf | 499.66 kB | Adobe PDF | View/Open |
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