Subnormalizers and p-elements in finite groups

The probability that two elements of a finite group commute (i.e. generate an abelian subgroup), also known as the extit{degree of commutativity of G}, dc(G), is a well studied topic in group theory. A first result involving this probability is the theorem by Gustafson which states that the degree of commutativity of a finite nonabelian group is less than 5/8. As a natural generalization, one can consider the probability that two elements of a finite group generate a nilpotent subgroup, the degree of nilpotence. A Gustafson-like theorem for the degree of nilpotence was proved by Guralnick and Wilson in 2000 using the classification of finite simple groups: the degree of nilpotence of a finite nonnilpotent group is less than 1/2. We give a classification-free proof of this theorem. In our proof, an important role is played by the probability sp(G), i.e., the arithmetic mean of the ratios between |S_G(x)| and |G| where S_G(x) is the Wielandt's subnormalizer of x. The main result for our proof is a probabilistic version of Wielandt's criterion for subnormality, which depends on a formula for the order of subnormalizers of p-subgroups proved by Casolo. The analysis of the probability sp(G) leads to the following theorem: if G is a nonsolvable group then sp(G) <1/6. The proof of this theorem involves the classification of finite simple groups and takes up a large portion of the thesis. Inspired by a problem related to this theorem, the final chapter of the thesis is devoted to the investigation of the ratio between the number of p-elements and the order of a Sylow p-subgroup in a finite group. For a p-solvable group G we prove that this ratio is greater than the (p-1)/p th power of the number of Sylow p-subgroups of G. As for the non-p-solvable case, we state a conjecture that, if true, would imply the aforementioned bound for any finite group G and we provide a reduction of the conjecture to finite almost simple groups.