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Suzuki, Michio Cobb, Philip Abram 2014-12-16T06:18:20Z 2014-12-16T06:18:20Z 10000-01-01 1987 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 $\pi$. Made available in DSpace on 2014-12-16T06:18:20Z (GMT). No. of bitstreams: 1 8803005.pdf: 2982933 bytes, checksum: ccc43598f315922f9b24e7bdaad86dbe (MD5) Previous issue date: 1987 Embargo set by: Seth Robbins for item 71426 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 63 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. http://hdl.handle.net/2142/71260 (UMI)AAI8803005 Mathematics Existence and Conjugacy of Hall Subgroups and Embedding of Pi-Subgroups text Ph.D. Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation