In their paper "Finite groups whose noncentral commuting elements have centralizers of equal size", S. Dolfi, M. Herzog and E. Jabara classify the groups in question- which they call $ CH$-groups- up to finite $p$-groups. Our goal is to investigate the finite $p$-groups in the class. The chief result is that a finite $p$-group that is a $ CH$-group either has an abelian maximal subgroup or is of class at most $p+1$. Detailed descriptions, in some cases characterisations up to isoclinism, are given.
M. Aschbacher (2001). Finite Group Theory. Cambridge University Press, Cambridge. S. Dolfi, M. Herzog and E. Jabara (2010). Finite groups whose noncentral commuting elements have centralizers of equal size. Bull. Aust. Math. Soc.. 82, 293-304 M. Hall (1959). The theory of groups. Macmillan, New York. B. Huppert (1967). Endliche Gruppen I. Springer, Heidelberg. I. M. Isaacs (2008). Subgroups generated by small classes in finite groups. Proc. Amer. Math. Soc.. 136, 2299-3301 A. Mann (1978). Conjugacy classes in finite groups. Israel J. Math.. 31, 78-84 A. Mann (2006). Elements of minimal breadth in finite $p$-groups and Lie algebras. J. Aust. Math. Soc.. 81, 209-214 G. Parmeggiani and B. Stellmacher (1999). $p$-groups of small breadth. J. Algebra. 213 (1), 52-68 L.Verardi (1987). Semiextraspecial groups of exponent $p$. Ann. Mat. Pura. Appl. (4). 148, 131-171
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