Shen's conjecture on groups with given same order type

TL;DR

This paper investigates the relationship between the prime divisors of a nilpotent group and its same-order type, providing partial answers to Shen's conjecture and examining groups with subgroups having limited same-order types.

Contribution

It proves that for nilpotent groups, the number of prime divisors is at most the size of the same-order type, and explores groups with subgroups having at most two same-order classes.

Findings

For nilpotent groups, |ta(G)|  |(G)|

Groups with all proper subgroups having ta  2 are characterized

Partial validation of Shen's conjecture for specific group classes

Abstract

For any group $G$, we define an equivalence relation $\thicksim$ as below: $$\forall \ g, h \in G \ \ g\thicksim h \Longleftrightarrow |g|=|h|$$ 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 whoseproper subgroups, say $H$ have $|\alpha{(H)}|\leq 2$.