A New Criterion for Finite Noncyclic Groups |
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Authors: | Wei Zhou Zeyong Duan |
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Institution: | College of Mathematics and Statistics , Southwest University , Chongqing, China |
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Abstract: | Let H be a subgroup of a group G. We say that H satisfies the power condition with respect to G, or H is a power subgroup of G, if there exists a non-negative integer m such that H = G m = 〈 g m |g ? G 〉. In this note, the following theorem is proved: Let G be a group and k the number of nonpower subgroups of G. Then (1) k = 0 if and only if G is a cyclic group (theorem of F. Szász); (2) 0 < k < ∞ if and only if G is a finite noncyclic group; (3) k = ∞ if and only if G is a infinte noncyclic group. Thus we get a new criterion for the finite noncyclic groups. |
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Keywords: | Cyclic group Dedekind group Power subgroup |
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