Volume 50, Issue 4
On Finite Groups Whose Nilpotentisers Are Nilpotent Subgroups

Seyyed Majid Jafarian Amiri & Hojjat Rostami

J. Math. Study, 50 (2017), pp. 307-313.

Published online: 2018-04

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  • Abstract

Let $G$ be a finite group and $x ∈ G.$ The nilpotentiser of $x$ in $G$ is defined to be the subset $Nil_G(x) =\{y∈ G :\langle x,y \rangle \ is\ nilpotent\}.$ $G$ is called an $\mathcal{N}$-group ($n$-group) if $Nil_G(x)$ is a subgroup (nilpotent subgroup) of $G$ for all $x ∈ G\setminus Z^∗(G)$ where $Z^∗(G)$ is the hypercenter of $G$. In the present paper, we determine finite $\mathcal{N}$-groups in which the centraliser of each noncentral element is abelian. Also we classify all finite $n$-groups.

  • AMS Subject Headings

20D60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

sm_jafarian@znu.ac.ir (Seyyed Majid Jafarian Amiri)

h.rostami5991@gmail .com (Hojjat Rostami)

  • BibTex
  • RIS
  • TXT
@Article{JMS-50-307, author = {Amiri , Seyyed Majid Jafarian and Rostami , Hojjat}, title = {On Finite Groups Whose Nilpotentisers Are Nilpotent Subgroups}, journal = {Journal of Mathematical Study}, year = {2018}, volume = {50}, number = {4}, pages = {307--313}, abstract = {

Let $G$ be a finite group and $x ∈ G.$ The nilpotentiser of $x$ in $G$ is defined to be the subset $Nil_G(x) =\{y∈ G :\langle x,y \rangle \ is\ nilpotent\}.$ $G$ is called an $\mathcal{N}$-group ($n$-group) if $Nil_G(x)$ is a subgroup (nilpotent subgroup) of $G$ for all $x ∈ G\setminus Z^∗(G)$ where $Z^∗(G)$ is the hypercenter of $G$. In the present paper, we determine finite $\mathcal{N}$-groups in which the centraliser of each noncentral element is abelian. Also we classify all finite $n$-groups.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v50n4.17.01}, url = {http://global-sci.org/intro/article_detail/jms/11319.html} }
TY - JOUR T1 - On Finite Groups Whose Nilpotentisers Are Nilpotent Subgroups AU - Amiri , Seyyed Majid Jafarian AU - Rostami , Hojjat JO - Journal of Mathematical Study VL - 4 SP - 307 EP - 313 PY - 2018 DA - 2018/04 SN - 50 DO - http://doi.org/10.4208/jms.v50n4.17.01 UR - https://global-sci.org/intro/article_detail/jms/11319.html KW - Finite group, nilpotentiser, $\mathcal{N}$-group. AB -

Let $G$ be a finite group and $x ∈ G.$ The nilpotentiser of $x$ in $G$ is defined to be the subset $Nil_G(x) =\{y∈ G :\langle x,y \rangle \ is\ nilpotent\}.$ $G$ is called an $\mathcal{N}$-group ($n$-group) if $Nil_G(x)$ is a subgroup (nilpotent subgroup) of $G$ for all $x ∈ G\setminus Z^∗(G)$ where $Z^∗(G)$ is the hypercenter of $G$. In the present paper, we determine finite $\mathcal{N}$-groups in which the centraliser of each noncentral element is abelian. Also we classify all finite $n$-groups.

Seyyed Majid Jafarian Amiri & Hojjat Rostami. (2019). On Finite Groups Whose Nilpotentisers Are Nilpotent Subgroups. Journal of Mathematical Study. 50 (4). 307-313. doi:10.4208/jms.v50n4.17.01
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