Existence and Conjugacy of Hall Subgroups and Embedding of Pi-Subgroups

63 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987.Let G be a finite group and let $\pi$ be a set of primes. A Hall $\pi$-subgroup H is a subgroup of G such that H is divisible by only primes in $\pi$ and (G: H) is divisible by no primes in $\pi$. This is also written as "G satisfies $E\sb\pi$." If in addition any two Hall $\pi$-subgroups are conjugate in G, then we say that G satisfies $C\sb\pi$. If G satisfies $C\sb\pi$ and any $\pi$-subgroup of G is contained in some Hall $\pi$-subgroup, then G satisfies $D\sb\pi$.Hall's theorem asserts that any solvable group satisfies $D\sb\pi$. This paper determines when certain non-solvable groups satisfy $E\sb\pi$, $C\sb\pi$, and $D\sb\pi$. The relationship between these statements is also explored. For example, it is shown, contrary to previous conjecture, that $E\sb\pi$ does not imply $C\sb\pi$, even if 2 $\not\in$ $\pi$. However, we see (in the main theorem) that $E\sb\pi$ implies $C\sb\pi$, at least if $\pi$ consists of sufficiently large primes.This statement is proved by reducing the problem to the simple groups and using the recent classification. The verification is done for simple groups first in the characteristic case, and then in the non-characteristic case for the classical groups. It is then shown that this proves the theorem to within a finite set of primes.This investigation also yields necessary and sufficient conditions for $E\sb\pi$ in groups of Lie type in the characteristic case. In the non-characteristic case, we examine two families of classical groups in the special case that $\pi$ contains only two primes. For the general linear group, necessary and sufficient conditions for $D\sb\pi$ are given. Turning to the orthogonal group, we determine exactly when $E\sb\pi$ holds if both primes in $\pi$ are odd.Finally, we can describe the general structure of odd-order Hall subgroups, at least in the classical groups. In the characteristic case, they are subgroups of a Borel subgroup. For the non-characteristic case, a Hall subgroup H has a normal homocyclic abelian subgroup N. Further, the index (G: N) is divisible only by the smallest prime in $\pi$.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD