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Projective special unitary groups $\mathrm{PSU}(5,q)$, where $$\frac{q^4-q^3+q^2-q+1}{(5,q+1)}$$ is a prime, is uniquely determined by its order and the size of one conjugacy class.

One of the important questions that remains after the classification of the finite simple groups is how to recognize a simple group via specific properties. For example, authors have been able to use graphs associated to element orders and to number of elements with specific orders to determine simple groups up to isomorphism. In this paper, we prove that Suzuki groups $Sz\left(q\right)$, where $q\pm \sqrt{2q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order.

We proved that the symplectic groups $\mathrm{PSp}(4,2^n)$, where $2^{2n}+1$ is a Fermat prime number is uniquely determined by its order, the first largest element orders and the second largest element orders.