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The breadth of an element $x$ in a finite $p$-group G is defined to be that integer $b = br(x)$ such that $p ^b = |G : C_ G (x)|$, while the breadth $br(G)$ of $G$ is the supremum of $\{br_ G (x) | x \in G\}$.

In [ J.A. Gallian, On the breadth of a finite p -group, Math. Z. 126 (1972), 224–226.], it has been proved that if $c$ denotes the nilpotency class of $G$, then $ c < 1+(pb−1)/(p−1)$. Does the above result work for all values of $p$ an $b$? (Because it seems that it does not work for $D_8$ and $Q_8$ )

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  • $\begingroup$ If you check the paper, you will find that the result has the extra hypothesis $b>1$, which does not hold in $D_8$ or $Q_8$. $\endgroup$
    – Derek Holt
    Sep 1, 2014 at 12:18
  • $\begingroup$ @ Prof. Holt, my try for finding the main paper failed and I find this result in the other paper which missed the fact that $b>1$. Is there any special result when $p=2$ and $b=1$. $\endgroup$
    – sara
    Sep 1, 2014 at 12:28
  • $\begingroup$ When $b=1$, the known result is $c < 1 + pb/(p-1)$, proved in: Leedham-Green, C., Neumann, P., Wiegold, J.: The breadth and the class of a finite p-group. J. London Math. Soc. (2), 1, 409-420 (1969). $\endgroup$
    – Derek Holt
    Sep 1, 2014 at 12:46

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