Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.01627 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929253726027776 |
|---|---|
| author | Ercan, Gülin Güloğlu, İsmail Ş. |
| author_facet | Ercan, Gülin Güloğlu, İsmail Ş. |
| contents | Let $A$ be a finite nilpotent group acting fixed point freely on the finite (solvable) group $G$ by automorphisms. It is conjectured that the nilpotent length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the order of $A$ counted with multiplicities. In the present paper we consider the case $A$ is cyclic and obtain that the nilpotent length of $G$ is at most $2\ell(A)$ if $|G|$ is odd. More generally we prove that the nilpotent length of $G$ is at most $2\ell(A)+ \mathbf{c}(G;A)$ when $G$ is of odd order and $A$ normalizes a Sylow system of $G$ where $\mathbf{c}(G;A)$ denotes the number of trivial $A$-modules appearing in an $A$-composition series of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01627 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Noncoprime action of a cyclic group Ercan, Gülin Güloğlu, İsmail Ş. Group Theory Let $A$ be a finite nilpotent group acting fixed point freely on the finite (solvable) group $G$ by automorphisms. It is conjectured that the nilpotent length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the order of $A$ counted with multiplicities. In the present paper we consider the case $A$ is cyclic and obtain that the nilpotent length of $G$ is at most $2\ell(A)$ if $|G|$ is odd. More generally we prove that the nilpotent length of $G$ is at most $2\ell(A)+ \mathbf{c}(G;A)$ when $G$ is of odd order and $A$ normalizes a Sylow system of $G$ where $\mathbf{c}(G;A)$ denotes the number of trivial $A$-modules appearing in an $A$-composition series of $G$. |
| title | Noncoprime action of a cyclic group |
| topic | Group Theory |
| url | https://arxiv.org/abs/2307.01627 |