Abstract
Let o(G) be the average order of a finite group G. We show that if o(G)<c, where c∈{136,114}, then G is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the set containing the average orders of all finite groups is not dense in [a,∞), for all a∈[0,136]. We also outline some results related to the integer values of the average order. Since group element orders is a popular research topic, we pose some open problems concerning the average order of a finite group throughout the paper.
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