Shen's conjecture on groups with given same order type
| Autor: | Leyli Jafari Taghvasani, Mohammad Zarrin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | International Journal of Group Theory, Vol 6, Iss 1, Pp 17-20 (2017) |
| Druh dokumentu: | article |
| ISSN: | 2251-7650 2251-7669 |
| DOI: | 10.22108/ijgt.2017.10631 |
| Popis: | For any group $G$, we define an equivalence relation $thicksim$ as below: [forall g, h in G gthicksim h Longleftrightarrow |g|=|h|] the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denote by $alpha{(G)}$. In this paper, we give a partial answer to a conjecture raised by Shen. In fact, we show that if $G$ is a nilpotent group, then $|pi(G)|leq |alpha{(G)}|$, where $pi(G)$ is the set of prime divisors of order of $G$. Also we investigate the groups all of whose proper subgroups, say $H$ have $|alpha{(H)}|leq 2$. |
| Databáze: | Directory of Open Access Journals |
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