Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images |
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Authors: | A Abdollahi A Faghihi A Mohammadi Hassanabadi |
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Institution: | 1. Department of Mathematics , University of Isfahan , Isfahan, Iran a.abdollahi@math.ui.ac.ir;3. Department of Mathematics , University of Isfahan , Isfahan, Iran |
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Abstract: | A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p 8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups. |
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Keywords: | Endomorphisms of groups 2-Engel Groups p-Groups Near-rings |
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