p-grupos finitos
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1997
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17-07-1997
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Abstract
In the first part, new bounds for the number of conjugacy classes of maximal length of a finite $p$-group $G$, $r(G)$, are obtained, and they are related with the length of these classes. If $r(G)=p^m-b-1$, there exists a unique normal subgroup of order $p^b$, $N_b$, which is characteristic, and structural properties of these groups are studied when $b\le 3$, by paying special attention to the relation between $N_b$, $Z(G)$ and $G$. In the second part, new bounds for the degree of commutativity of a $p$-group of maximal class are established. In Chapter 2, the bounds known for $c$ are reviewed. In Chapter 3, the reuslts obtained by Shepherd for $c_0\le 4$ are extended to $c_0\le 10$ by means of the development of new computational techniques. In Chapter 4, we present some tables which give the bounds for $c$ as a function of $c_0$ and $l$ for $p\le 3$, obtained with the help of the mentioned computational techniques, and the finest possible bounds for $c$ for the majority of the values of $c_0$ and $l$ and every prime $p$ are conjectured. The majority of these conjectures are solved, and hence the bounds given by Shepherd, Leedham-Green and McKay, and Fernández-Alcober are improved. It is also shown the optimality of these bounds by means of the construction of the Lie algebras for which these bounds are attained.