“Element orders” is one of the most fundamental concepts in group theory. It plays an important role in research in group theory, which can be seen from the famous Burnside problem. Some well-known group theory specialists, such as B.H. Neumann, G. Higman, M. Suzuki and others, have studied the groups whose element orders are of the special values (see [1, 2, 3]). In 1981 Shi investigated the finite groups all of whose elements are of prime order except the identity element, and got the interesting result: The alternating group A5 can be characterized only by its element orders (see [4]). The above work was repeated in [5] since [4] was published in Chinese and not reviewed in “Mathematical Reviews”.

Let G be a group. Denote by πe(G) the set of all orders of elements in G. Obviously, πe(G) is a subset of the set Z+ of positive integers, and it is closed and partially ordered under divisibility. The converse problem, which divisibility closed subsets of Z+ can be the sets of element orders of groups, is more difficult. Let Г be a subset of Z+ and h(Г) be the number of isomorphism classes of groups G such that πe(G) = Г. For a given Г, groups G such that πe(G) = Г do not necessarily exist. However, for a given group G, we have h(πe(G)) ≥ 1.