Abstract
Let G be a finite group and ω(G) be the set of element orders of G. Let k∈ω(G) and mk be the number of elements of order k in G. Let nse(G)={mk|k∈ω(G)}. In this paper, we prove that if G is a finite group such that nse(G) = nse(H), where H=PSU(3,3) or PSL(3,3), then G≅H.
Highlights
We devote this section to relevant definitions, basic facts about nse, and a brief history of this problem
We express by π ( G ) the set of prime divisors of | G |, and by ω ( G ), we introduce the set of order of elements from G
By Lemma 1, we can assume that G is finite
Summary
We devote this section to relevant definitions, basic facts about nse, and a brief history of this problem. Characterization of a group G by nse(G) and | G |, for short, deals with the number of elements of order k in the group G and | G |, where one must answer the question “is a finite group G, can be characterized by the set nse(G) and | G |?” While mathematicians might undoubtedly give many answers to such a question, the answer in Shao et al [2,3] would probably rank near the top of most responses They proved that if G is a simple k i (i = 3, 4) group, G is characterizable by nse( G ).
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