# The Influence of some embedding properties of subgroups on the structure of a finite group

2018
Kızmaz, Muhammet Yasir
In a finite group \$G\$, a subgroup \$H\$ is called a \$TI\$-subgroup if \$H\$ intersects trivially with distinct conjugates of itself. Suppose that \$H\$ is a Hall \$pi\$-subgroup of \$G\$ which is also a \$TI\$-subgroup. A famous theorem of Frobenius states that \$G\$ has a normal \$pi\$-complement whenever \$H\$ is self normalizing. In this case, \$H\$ is called a Frobenius complement and \$G\$ is said to be a Frobenius group. A first main result in this thesis is the following generalization of Frobenius' Theorem. textbf{Theorem.}textit{ Let \$H\$ be a \$TI\$-subgroup of \$G\$ which is also a Hall subgroup of \$N_G(H)\$. Then \$H\$ has a normal complement in \$N_G(H)\$ if and only if \$H\$ has a normal complement in \$G\$. Moreover, if \$H\$ is nonnormal in \$G\$ and \$H\$ has a normal complement in \$N_G(H)\$ then \$H\$ is a Frobenius complement.} In the above configuration, the group \$G\$ need not be a Frobenius group, but the second part of the theorem guarantees the existence of a Frobenius group into which \$H\$ can be embedded as a Frobenius complement. Another contribution of this thesis is the following theorem, which extends a result of Gow (see Theorem ref{int gow}) to \$pi\$-separable groups. This result shows that the structure of a \$pi\$-separable group admitting a Hall \$pi\$-subgroup which is also a \$TI\$-subgroup is very restricted. textbf{Theorem.}textit{ Let \$H\$ be a nonnormal \$TI\$-subgroup of the \$pi\$-separable group \$G\$ where \$pi\$ is the set of primes dividing the order of \$H\$. Further assume that \$H\$ is a Hall subgroup of \$N_G(H)\$. Then the following hold:} textit{\$a)\$ \$G\$ has \$pi\$-length \$1\$ where \$G=O_{pi'}(G)N_G(H)\$;} textit{\$b)\$ there is an \$H\$-invariant section of \$G\$ on which the action of \$H\$ is Frobenius. This section can be chosen as a chief factor of \$G\$ whenever \$O_{pi'}(G)\$ is solvable;} textit{\$c)\$ \$G\$ is solvable if and only if \$O_{pi'}(G)\$ is solvable and \$H\$ does not involve a subgroup isomorphic to \$SL(2,5)\$.} In the last chapter we focus on giving alternative proofs without character theory for the following two solvability theorems due to Isaacs (cite{isa3}, Theorem 1 and Theorem 2). Our proofs depend on transfer theory and graph theory. textbf{Theorem.} textit{Let \$G\$ be a finite group having a cyclic Sylow \$p\$-subgroup. Assume that every \$p'\$-subgroup of \$G\$ is abelian. Then \$G\$ is either \$p\$-nilpotent or \$p\$-closed.} textbf{Theorem.} textit{Let G be a finite group and let \$pneq 2\$ and \$q\$ be primes dividing \$

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Citation Formats
M. Y. Kızmaz, “The Influence of some embedding properties of subgroups on the structure of a finite group,” Ph.D. - Doctoral Program, Middle East Technical University, 2018. 