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Zbl 0587.20012
Vera Lopez, Antonio
The number of conjugacy classes in a finite nilpotent group. (English)
[J] Rend. Sem. Mat. Univ. Padova 73, 209-216 (1985). ISSN 0041-8994

Let G be a finite group of order $p\sp m$ (p prime) with centre Z(G) of order $p\sp b$ and let r(G) be the number of conjugacy classes of G. This paper contains a number of interesting equations, inequalities and congruences relating r(G) to other invariants of G. A principal and representative result is the following one. Suppose G has a maximal abelian subgroup A of order $p\sp a$. Then there exists an integer $k\ge 0$ such that $$ r(G)=(p\sp{2a}/p\sp m)+(p\sp b(p+1)(p\sp{m-a}-1)/p\sp{m- a})+k(p\sp 2-1)(p-1)/p\sp{m-a}.$$
[T.J.Laffey]

MSC 2000:: *20D15 Nilpotent finite groups
20D60 Arithmetic and combinatorial problems on finite groups

Keywords: finite p-group; centre; number of conjugacy classes; maximal abelian subgroup

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