Thompson's conjecture for the alternating group of degree 2p2p and 2p+12p+1

summary:For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N(G) = N(L)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z(G)=1$ and $N(G) = N(A_{i})$ is necessarily isomorphic to $A_{i}$, where $i\in \{2p,2p+1\}$