2,974 research outputs found
A bound on element orders in the holomorph of a finite group
Let be a finite group. We prove a theorem implying that the orders of
elements of the holomorph are bounded from above by
, 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 -groups of order
first studied by Jonah and Konvisser in 1975 as examples for finite
-groups with abelian automorphism group, and we show some necessary
conditions for a finite -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 , examples of finite -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 -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 be a finite group, and assume that has an automorphism of order
at least , with . Generalizing recent
analogous results of the author on finite groups with a large automorphism
cycle length, we prove that if , then is abelian, and if
, then is solvable, whereas in general, the assumption implies
, where
denotes the solvable radical of . 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 assumed to
admit an -orbit of length at least , for a
given . Our main results are the following: The orders
of the nonabelian composition factors of are then bounded in terms of
, and if , then is solvable. On the other hand,
for each nonabelian finite simple group , there is a constant
such that occurs with arbitrarily large
multiplicity as a composition factor in some finite group having an
-orbit of length at least .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 of a finite nilpotent group has a cycle whose
length coincides with . Also, we give two new sufficient
conditions for an automorphism of an arbitrary finite group to satisfy
this property, namely when is a product of at most two
prime powers or when 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 multiplicity-bounding if and only if a finite group
on which the word map of has a fiber of positive proportion can only
contain each nonabelian finite simple group as a composition factor with
multiplicity bounded in terms of and . 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 , where is a
single variable and 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 , if a finite group
has an automorphism with a cycle of length at least , then
the index of the solvable radical in is bounded
from above in terms of , and such a condition is strong enough to imply
solvability of if and only if . Furthermore,
considering, for exponents , the condition that a finite
group have an automorphism with a cycle of length at least , such a
condition is strong enough to imply for
if and only if . 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 for and some superlinear and subcritical nonlinearity
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
- β¦