A Note on Shen’s Conjecture on Groups with Given Same-Order Type

Abstract

Let $G$ be a group. Define an equivalence relation $\sim$ on $G$ as follows: for $x,y \in G$, $x \sim y$ if $x$ and $y$ have same order. The set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$. Let $s_{k}(G)$ and $\pi_{e}(G)$ denote the number of elements of order $k$ and the set of element orders of the finite group $G$, respectively. Shen (2012) posed the following conjecture: let $G$ be a group of order $p^{l}$ with same-order type $\{1,m,n\}$, and let $|\pi_{e}(G)|>3$. If $p=2$ and $s_{2^{i}}(G)\neq0$ for $i\ge2$, then $s_{2^{i}}(G)=2^{l-2}$. If $p>2$, then there is no such group. In this paper, we give a partial answer to this conjecture. In fact, for $p=2$ with a counterexample, we give negative answer to the above conjecture, and for $p>2$, we find that above conjecture holds for finite $p$-groups of nilpotency class less than $p$.