On the generating graphs of the symmetric and alternating groups

Erdem, Fuat
Dixon showed that the probability that a random pair of elements in the symmetric group $S_n$ generates $S_n$ or the alternating group $A_n$ tends to $1$ as $n to infty$. (A generalization of this result was given by Babai and Hayes.) The generating graph $Gamma(G)$ of a finite group $G$ is defined to be the simple graph on the set of non-identity elements of $G$ with the property that two elements are connected by and edge if and only if they generate $G$. The purpose of this thesis is to study the graphs $Gamma(S_n)$ and $Gamma(A_n)$. We prove that the graphs $Gamma(S_n)$ and $Gamma(A_n)$ contain Hamiltonian cycles provided that $n geq 107$. This improves a recent result of Breuer, Guralnick, Lucchini, Mar'oti and Nagy. Our result can be viewed as another step towards the conjecture of Breuer, Guralnick, Lucchini, Mar'oti and Nagy stating that for an arbitary finite group $G$ of order at least $4$ the generating graph $Gamma(G)$ contains a Hamiltonian cycle if and only if $G/N$ is cyclic for every non-trivial normal subgroup $N$ of $G$. (This is a stronger form of an older conjecture of Breuer, Guralnick and Kantor.) Our results may have applications to dimensions of fixed point spaces of elements of a finite group $G$ acting on a finite dimensional vector space $V$ with $C_{V}(G) = 0$.


Approximate Pseudospin and Spin Solutions of the Dirac Equation for a Class of Exponential Potentials
Arda, Altug; Sever, Ramazan; TEZCAN, CEVDET (2010-02-01)
The Dirac equation is solved for some exponential potentials the hypergeometric-type potential, the generalized Morse potential, and the Poschl-Teller potential with any spin-orbit quantum number kappa in the case of spin and pseudospin symmetry. We have approximated for non s-waves the centrifugal term by an exponential form. The energy eigenvalue equations and the corresponding wave functions are obtained by using a generalization of the Nikiforov-Uvarov method.
On a Fitting length conjecture without the coprimeness condition
Ercan, Gülin (Springer Science and Business Media LLC, 2012-08-01)
Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p (n) q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.
On the notion of stability of order convergence in vector lattices
Gorokhova, SG; Emelyanov, Eduard (1994-09-01)
In the theory of Banach lattices the following criterion for a norm to be order continuous is established: a norm is order continuous if and only if every order bounded sequence of positive pairwise disjoint elements in a lattice converges to zero in norm. In this paper we give a criterion for order convergence to be stable in a rather wide class of vector lattices which includes all Köthe spaces. The formulation of the criterion is analogous to that of the above-mentioned criterion for a norm to be order c...
On the nilpotent length of a finite group with a frobenius group of automorphisms
Öğüt, Elif; Ercan, Gülin; Güloğlu, İsmail Ş.; Department of Mathematics (2013)
Let G be a finite group admitting a Frobenius group FH of automorphisms with kernel F and complement H. Assume that the order of G and FH are relatively prime and H acts regularly on the fixed point subgroup of F in G. It is proved in this thesis that the nilpotent length of G is less than or equal to the sum of the nilpotent length of the commutator group of G and F with 1 and the nilpotent length of the commutator group of G and F is equal to the nilpotent length of the fixed point subgroup of H in the co...
On the influence of fixed point free nilpotent automorphism groups
Ercan, Gülin (2017-12-01)
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 for all nonidentity elements . Let FH be a Frobenius-like group with complement H of prime order such that is of prime order. Suppose that FH acts on a finite group G by automorphisms where in such a way that In the present paper we prove that the Fitting series of coincides with the intersections of with the Fitting series of G, and the nilpotent length of G exceeds the...
Citation Formats
F. Erdem, “On the generating graphs of the symmetric and alternating groups,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.