A bound on element orders in the holomorph of a finite group

Let $G$ be a finite group. We prove a theorem implying that the orders of elements of the holomorph $\operatorname{Hol}(G)$ are bounded from above by $|G|$, and we discuss an application to bounding automorphism orders of finite groups.Comment: 5 page

On the endomorphism monoids of some groups with abelian automorphism group

We investigate the endomorphism monoids of certain finite $p$-groups of order $p^8$ first studied by Jonah and Konvisser in 1975 as examples for finite $p$-groups with abelian automorphism group, and we show some necessary conditions for a finite $p$-group to have commutative endomorphism monoid. As a by-product, apart from formulas for the number of conjugacy classes of endomorphisms of said groups, we will be able to derive the following: There exist nonabelian groups with commutative endomorphism monoid, and having commutative endomorphism monoid is a group property strictly stronger than having abelian automorphism group. Furthermore, using a result of Curran, this will enable us to give, for all primes $p$, examples of finite $p$-groups which are direct products and have abelian automorphism group.Comment: 17 page

On the dynamics of endomorphisms of finite groups

Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting's Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed $1$-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hern\'andez-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their "ramification behavior". Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hern\'andez-Toledo on primary cyclic groups of odd order.Comment: 8 page

Finite groups with an automorphism of large order

Let $G$ be a finite group, and assume that $G$ has an automorphism of order at least $\rho|G|$, with $\rho\in\left(0,1\right)$. Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we prove that if $\rho>1/2$, then $G$ is abelian, and if $\rho>1/10$, then $G$ is solvable, whereas in general, the assumption implies $[G:\operatorname{Rad}(G)]\leq\rho^{-1.78}$, where $\operatorname{Rad}(G)$ denotes the solvable radical of $G$. Furthermore, we generalize an example of Horo\v{s}evski\u{\i} to show that in finite groups, the quotient of the maximum automorphism order by the maximum automorphism cycle length may be arbitrarily large.Comment: 9 page

Finite groups with a large automorphism orbit

We study the nonabelian composition factors of a finite group $G$ assumed to admit an $\operatorname{Aut}(G)$-orbit of length at least $\rho|G|$, for a given $\rho\in\left(0,1\right]$. Our main results are the following: The orders of the nonabelian composition factors of $G$ are then bounded in terms of $\rho$, and if $\rho>\frac{18}{19}$, then $G$ is solvable. On the other hand, for each nonabelian finite simple group $S$, there is a constant $c(S)\in\left(0,1\right]$ such that $S$ occurs with arbitrarily large multiplicity as a composition factor in some finite group $G$ having an $\operatorname{Aut}(G)$-orbit of length at least $c(S)|G|$.Comment: 35 page

On finite groups where the order of every automorphism is a cycle length

Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horo\v{s}evski\u{i}'s result that every automorphism $\alpha$ of a finite nilpotent group has a cycle whose length coincides with $\mathrm{ord}(\alpha)$. Also, we give two new sufficient conditions for an automorphism $\alpha$ of an arbitrary finite group to satisfy this property, namely when $\mathrm{ord}(\alpha)$ is a product of at most two prime powers or when $\alpha$ has a sufficiently large cycle. This will allow us to show that the least order of a group where this property is violated is 120. Finally, we observe that any finite group embeds both into a group with this property (as all finite symmetric groups enjoy the property) as well as into a finite group not having this property.Comment: 16 page

Fibers of word maps and the multiplicities of nonabelian composition factors

Call a reduced word $w$ multiplicity-bounding if and only if a finite group on which the word map of $w$ has a fiber of positive proportion $\rho$ can only contain each nonabelian finite simple group $S$ as a composition factor with multiplicity bounded in terms of $\rho$ and $S$. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form $x^e$, where $x$ is a single variable and $e$ an odd integer.Comment: 28 pages, 1 table; v2: some revisions necessitated by the author's discovery that the power word x^8 is NOT multiplicity-bounding (which was originally overlooked because of a programming error

Cycle lengths in finite groups and the size of the solvable radical

We prove the following: For any $\rho\in\left(0,1\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $\rho\cdot|G|$, then the index of the solvable radical $\operatorname{Rad}(G)$ in $G$ is bounded from above in terms of $\rho$, and such a condition is strong enough to imply solvability of $G$ if and only if $\rho>\frac{1}{10}$. Furthermore, considering, for exponents $e\in\left(0,1\right)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G|^e$, such a condition is strong enough to imply $|\operatorname{Rad}(G)|\to\infty$ for $|G|\to\infty$ if and only if $e>\frac{1}{3}$. We also prove similar results for a larger class of bijective self-transformations of finite groups, so-called periodic affine maps.Comment: 20 pages, 1 table, complete revision of the old version (from Jan 2015

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian $(-\Delta)^{\alpha/2}$ for $\alpha \in\left( 1,2\right)$ and some superlinear and subcritical nonlinearity $G_{z}$ provided with a nonhomogeneous Dirichlet exterior boundary condition. Some sufficient conditions under which the set of weak solutions to the boundary value problem is nonempty and depends continuously in the Painleve-Kuratowski sense on distributed parameters and exterior boundary data are stated. The proofs of the existence results rely on the Mountain Pass Theorem. The application of the continuity results to some optimal control problem is also provided.Comment: 20 page

Optimal control of nonlinear systems governed by Dirichlet fractional Laplacian in the minimax framework

We consider an optimal control problem governed by a class of boundary value problem with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes is stated. The proof of the main result relies on variational structure of the problem. To show that boundary value problem with the Dirichlet fractional Laplacian has a weak solution we employ the renowned Ky Fan Theorem.Comment: 15 page