# Prime graphs of solvable groups

2020-8
Ulvi , Muhammed İkbal
If \$G\$ is a finite group, its prime graph \$Gamma_G\$ is constructed as follows: the vertices are the primes dividing the order of \$G\$, two vertices \$p\$ and \$q\$ are joined by an edge if and only if \$G\$ contains an element of order \$pq\$. This thesis is mainly a survey that gives some important results on the prime graphs of solvable groups by presenting their proofs in full detail.

# Suggestions

 NILPOTENT LENGTH OF A FINITE SOLVABLE GROUP WITH A FROBENIUS GROUP OF AUTOMORPHISMS Ercan, Gülin; Ogut, Elif (2014-01-01) We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C-CG(F) (h) = 1 for all nonidentity elements h is an element of H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.
 Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms Ercan, Gülin; Khukhro, E. I. (2014-07-01) A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with certain additional restrictions (which are satisfied, in particular, if either |FH| is odd or |H| = 2). In the case where G is a finite p-group such that G = [G, F] it is proved that the rank of G is bounded ab...
 Frobenius-like groups as groups of automorphisms Ercan, Gülin; Khukhro, Evgeny (2014-01-01) A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F, F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms FH aiming at restrictions on G in terms of C-G(H) and focusing mainly on bounds for the Fitting height and rela...
 Frobenius groups of automorphisms with almost fixed point free kernel Ercan, Gülin (2019-03-01) Let FH be a Frobenius group with kernel F and complement H, acting coprimely on the finite solvable group G by automorphisms. We prove that if C-G(H) is of Fitting length n then the index of the n-th Fitting subgroup F-n(G) in G is bounded in terms of vertical bar C-G(F)vertical bar and vertical bar F vertical bar. This generalizes a result of Khukhro and Makarenko  which handles the case n = 1.
 On local finiteness of periodic residually finite groups Kuzucouoglu, M; Shumyatsky, P (2002-10-01) Let G be a periodic residually finite group containing a nilpotent subgroup A such that C-G (A) is finite. We show that if [A, A(g)] is finite for any g is an element of G, then G is locally finite.
Citation Formats
M. İ. Ulvi, “Prime graphs of solvable groups,” M.S. - Master of Science, Middle East Technical University, 2020. 