Let $ G $ be a finite group and $ \\psi(G)=\\sum_{x \\in G} |x| $ , where $ |x| $ is the order of the element $ x $ . More generally , if $ X $ is a subset of $ G $ , then $ \\psi(X) $ denotes the sum of the orders of all elements of $ X $.For example , $ \\psi(C_4) = 1 #43;2 #43;4 #43;4 = 11 $ and $ \\psi(C_2 \imes C_2) = 1 #43;2 #43;2 #43;2 = 7$ , where $ C_n $ is a cyclic group of order $ n $ . Amiri and Jafarian Amiri asked what information about $ G $ can be recovered if they know both $ \\psi(G) $ and $ |G| $ . They showed that if $ G $ is non-cyclic of order $n$ , then $ \\psi(G) lt; \\psi(C_n) $ and $ \\psi(G) = \\psi(C_n) $ if and only if $ G \\cong C_n $ . Thus the sum of element orders of $ C_n $ is bigger than that of any other group of order $ n$ . It follows that for each positive integer $n$ , the cyclic group of order $n$ is uniquely determined up to isomorphism by its order and the sum of the orders of its elements . In general , however , the invariants $ |G| $ and $ \\psi(G) $ do not determine $G$ , sometimes $ \\psi(G) $ determines $G$ up to isomorphism even without knowing $ |G| $ .
