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FINITE GROUPS WHOSE NONCENTRAL COMMUTING ELEMENTS HAVE CENTRALIZERS OF EQUAL SIZE

  • SILVIO DOLFI (a1), MARCEL HERZOG (a2) and ENRICO JABARA (a3)

Abstract

A finite group is called a CH-group if for every x,yGZ(G), xy=yx implies that . Applying results of Schmidt [‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova44 (1970), 97–131] and Rebmann [‘F-Gruppen’, Arch. Math. 22 (1971), 225–230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups.

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Copyright

Corresponding author

For correspondence; e-mail: herzogm@post.tau.ac.il

Footnotes

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The first and the third authors were partially supported by the MIUR project ‘Teoria dei gruppi e applicazioni’.

Footnotes

References

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[1] GAP– Groups, Algorithms and Programming, Version 4.4.12 (2008), http://www.gap-system.org.
[2]Isaacs, I. M., ‘Subgroups generated by small classes in finite groups’, Proc. Amer. Math. Soc. 136 (2008), 22992301.
[3]Ishikawa, K., ‘On finite p-groups which have only two conjugacy lengths’, Israel J. Math. 129 (2002), 119123.
[4]Ito, N., ‘On finite groups with given conjugate type, I’, Nagoya J. Math. 6 (1953), 1728.
[5]Mann, A., ‘Conjugacy classes in finite groups’, Israel J. Math. 31 (1978), 7884.
[6]Mann, A., ‘Elements of minimal breadth in finite p-groups and Lie algebras’, J. Aust. Math. Soc. 81 (2006), 209214.
[7]Rebmann, J., ‘F-Gruppen’, Arch. Math. 22 (1971), 225230.
[8]Robinson, D. J. S., Finiteness Conditions and General Soluble Groups, Part 2 (Springer, Berlin, 1972).
[9]Robinson, D. J. S., A Course in the Theory of Groups (Springer, Berlin, 1982).
[10]Schmidt, R., ‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova 44 (1970), 97131.
[11]Schmidt, R., Subgroup Lattices of Groups (De Gruyter, Berlin, 1994).
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