## Hokkaido Mathematical Journal

### On the number of conjugacy classes in a finite p-group

Antonio VERA-LÓPEZ

#### Abstract

Let $G$ be a finite $p$-group of order $p^{m}=p^{2n+e}$, with $n$ a non-negative integer, $p$ a prime number and $e=0$ or $1$, and let $r(G)$ be the number of conjugacy classes of elements of $G$. Then the following equality, due to P. Hall, holds ([4], p. 549) : $$r(G)=(p^{2}-1)n+p^{e}+k(p^{2}-1)(p-1) ,$$ For some non-negative integer $k$. In this paper, we obtain new properties relative to $r(G)$ by the analysis of the number $r_{G}(gN)$ of conjugacy classes of elements of $G$ that intersect the coset $gN$, where $N$ is a normal subgroup of $G$ and $g$ any element of $G$. It contains a number of equations and congruences relating $r(G)$ to other invariants of $G$. In particular, our results improve the above equality of P. Hall, when $G$ has maximal nilpotent class or $n\leq p+1$. Examples are given, which make our improvements evident.

#### Article information

Source
Hokkaido Math. J., Volume 18, Number 3 (1989), 477-485.

Dates
First available in Project Euclid: 11 October 2013

https://projecteuclid.org/euclid.hokmj/1381517733

Digital Object Identifier
doi:10.14492/hokmj/1381517733

Mathematical Reviews number (MathSciNet)
MR1023229

Zentralblatt MATH identifier
0688.20009

#### Citation

VERA-LÓPEZ, Antonio. On the number of conjugacy classes in a finite p-group. Hokkaido Math. J. 18 (1989), no. 3, 477--485. doi:10.14492/hokmj/1381517733. https://projecteuclid.org/euclid.hokmj/1381517733