The Probability That an Ordered Pair of Elements is an Engel Pair

Hojjat Rostami

Abstract


Let G be a nite group. We denote by ep(G) the probability that
[x;n y] = 1 for two randomly chosen elements x and y of G and some posi-
tive integer n. For x 2 G we denote by EG(x) the subset fy 2 G : [y;n x] =
1 for some integer ng. G is called an E-group if EG(x) is a subgroup of G for all
x 2 G. Among other results, we prove that if G is an non-abelian E-group with
ep(G) > 1
6 , then G is not simple and minimal non-solvable.

Keywords


nite group, E-group, Engel element.

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DOI: https://doi.org/10.22342/jims.25.2.693.121-127

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Journal of the Indonesian Mathematical Society
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