A relation between infinite subsets and exterior center in groups

Author
Damghan University
Abstract
Let G be a group. Neumann to answer a question of Paul Erdos proved that every infinite subset of G has two different comuting elements if and only if G is center-by-finite. In this paper, we deal with Erdoschr('39')s question in different aspect and we show that every infinite subset X of G has two different elements x and y such that x^y=1 if and only if the exterior center of G ihas finite index.
Keywords

1. Abdollahi A., ” Finitely generated soluble groups with an Engel condition on infinite subsets”, Rend. Sem. Mat. Univ. Padova, 103 (2000) 47-49.

2. Abdollahi A., Taeri B., “A condition on finitely generated soluble groups”, Comm. Algebra, 27 (1999) 5633-5638.

3. Delizia C., Nicotera C., ”Groups with conditions on infinite subsets”, Ischia Group Theory 2006: Proceedings of a Conference in Honor of Akbar Rhmetulla, 4655, World Scientific Publishing, Singapore (2007).

4. Faramarzi Salles A., “Finitely generated soluble groups with a condition on infinite subsets”, Bull. Aust. Math. Soc., 87 (2013) 152-157.

5. Faramarzi Salles A., Pazandeh Shanbehbazari F., “ Locally graded groups with a condition on infinite subsets”, Inter. J. Group theory, Vol. 7 No. 4 (2018) 1-7.

6. Lennox, J.C., Wiegold J., “Extensions of a problem of Paul Erdös on groups”, J. Austral. Math. Soc. Ser. A, 31 (1981) 459-463.

7. McDermott A., “The nonabelian tensor product of groups: computations and structural results”, PhD thesis, National Univ. of Ireland, Galway, February (1998).

8. Neumann B. H., “A problem of Paul Erdös on groups”, J. Austral. Math. Soc., Ser. A 21 (1976) 467-472.

9. Niroomand P., Russo F., “A note on the exterior centralizer”, Arch. Math. 93 (2009) 505– 512.