scholarly journals FINITE SELF-SIMILAR p-GROUPS WITH ABELIAN FIRST LEVEL STABILIZERS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 355-364 ◽  
Author(s):  
ZORAN ŠUNIĆ
Keyword(s):  
Cyclic Group ◽  
Abelian Subgroup ◽  
Rooted Tree ◽  
Split Extension ◽  
Similar Action ◽  
Self Similar

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p. The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.

scholarly journals On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Francesco Noseda ◽  
Ilir Snopce
Keyword(s):  
Rooted Tree ◽  
Similarity Index ◽  
The Self ◽  
Self Similarity ◽  
Similar Action ◽  
3 Dimensional ◽  
Self Similar

Abstract Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

SCALING OF FLOW DISTANCE IN RANDOM SELF-SIMILAR CHANNEL NETWORKS

Fractals ◽  
2005 ◽  
Vol 13 (04) ◽  
pp. 265-282 ◽  
Author(s):  
BRENT M. TROUTMAN
Keyword(s):  
Drainage Basin ◽  
Empirical Studies ◽  
Rooted Tree ◽  
Network Size ◽  
Channel Networks ◽  
Tree Graphs ◽  
Affine Structures ◽  
Self Similar ◽  
Function Number ◽  
Behavior Characteristic

Natural river channel networks have been shown in empirical studies to exhibit power-law scaling behavior characteristic of self-similar and self-affine structures. Of particular interest is to describe how the distribution of distance to the outlet changes as a function of network size. In this paper, networks are modeled as random self-similar rooted tree graphs and scaling of distance to the root is studied using methods in stochastic branching theory. In particular, the asymptotic expectation of the width function (number of nodes as a function of distance to the outlet) is derived under conditions on the replacement generators. It is demonstrated further that the branching number describing rate of growth of node distance to the outlet is identical to the length ratio under a Horton-Strahler ordering scheme as order gets large, again under certain restrictions on the generators. These results are discussed in relation to drainage basin allometry and an application to an actual drainage network is presented.

Galois groups of doubly even octic polynomials

2019 ◽  
Vol 19 (01) ◽  
pp. 2050014
Author(s):  
Anna Altmann ◽  
Chad Awtrey ◽  
Sam Cryan ◽  
Kiley Shannon ◽  
Madeleine Touchette
Keyword(s):  
Conjugacy Class ◽  
Galois Group ◽  
Number Field ◽  
Cyclic Group ◽  
Dihedral Group ◽  
Irreducible Polynomial ◽  
Galois Groups ◽  
General Knowledge ◽  
Split Extension

Let [Formula: see text] be an irreducible polynomial with rational coefficients, [Formula: see text] the number field defined by [Formula: see text], and [Formula: see text] the Galois group of [Formula: see text]. Let [Formula: see text], and let [Formula: see text] be the Galois group of [Formula: see text]. We investigate the extent to which knowledge of the conjugacy class of [Formula: see text] in [Formula: see text] determines the conjugacy class of [Formula: see text] in [Formula: see text]. We show that, in general, knowledge of [Formula: see text] does not automatically determine [Formula: see text], except when [Formula: see text] is isomorphic to [Formula: see text] (the cyclic group of order 4). In this case, we show [Formula: see text] is isomorphic to a non-split extension of [Formula: see text] (the dihedral group of order 8) by [Formula: see text]. We also show that [Formula: see text] is completely determined when [Formula: see text] is isomorphic to [Formula: see text] and [Formula: see text] is a perfect square. In this case, [Formula: see text].

scholarly journals On connected transversals to Abelian subgroups

1994 ◽  
Vol 49 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Markku Niemenmaa ◽  
Tomas Kepka
Keyword(s):  
Cyclic Group ◽  
Abelian Subgroup ◽  
Loop Theory ◽  
Special Cases ◽  
Inner Mapping Group ◽  
Abelian Subgroups ◽  
Mapping Group ◽  
Finite Loop

In this paper we investigate the situation where a group G has an abelian subgroup H with connected transversals. We show that if H is finite then G is solvable. We also investigate some special cases where the structure of H is very close to the structure of a cyclic group. Finally we apply our results to loop theory and we show that if the inner mapping group of a finite loop Q is abelian then Q is centrally nilpotent.

scholarly journals Self-similar action potential cycle-to-cycle variability of Ca2+ and current oscillators in cardiac pacemaker cells

2020 ◽  
Author(s):  
Dongmei Yang ◽  
Alexey E. Lyashkov ◽  
Christopher H. Morrell ◽  
Ihor Zahanich ◽  
Yael Yaniv ◽  
...  
Keyword(s):  
Action Potential ◽  
Cardiac Pacemaker ◽  
Principal Component ◽  
Pacemaker Cells ◽  
Receptor Stimulation ◽  
Similar Action ◽  
Decay Times ◽  
Self Similar ◽  
The Mean ◽  
Kinetics Of

AbstractVariability of heart pacemaker cell action potential (AP) firing intervals (APFI) means that pacemaker mechanisms do not achieve equilibrium during AP firing. We tested whether mechanisms that underlie APFI, in rabbit sinoatrial cells are self-similar within and across the physiologic range of APFIs effected by autonomic receptor stimulation. Principal Component Analyses demonstrated that means and variabilities of APFIs and local Ca2+ releases kinetics, of AP induced Ca2+-transient decay times, of diastolic membrane depolarization rates, of AP repolarization times, of simulated ion current amplitudes, are self-similar across the broad range of APFIs (264 to 786 ms). Further, distributions of both mean APFIs and mean Ca2+ and membrane potential dependent coupled-clock function kinetics manifested similar power law behaviors across the physiologic range of mean APFIs. Thus, self-similar variability of clock functions intrinsic to heart pacemaker cells determines both the mean APFI and its interval variability, and vice versa.

scholarly journals Self-similar $k$-Graph C$^\ast $-Algebras

2019 ◽  
Author(s):  
Hui Li ◽  
Dilian Yang
Keyword(s):  
Recent Work ◽  
Product System ◽  
Similar Action ◽  
Self Similar ◽  
Stably Finite ◽  
Universal Coefficient Theorem

Abstract In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda $ and associate it a universal C$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda $ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

scholarly journals ON FINITE GENERATION OF SELF-SIMILAR GROUPS OF FINITE TYPE

2013 ◽  
Vol 23 (01) ◽  
pp. 69-79 ◽  
Author(s):  
IEVGEN V. BONDARENKO ◽  
IGOR O. SAMOILOVYCH
Keyword(s):  
Finite Group ◽  
Finite Type ◽  
Rooted Tree ◽  
Profinite Group ◽  
Similar Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Binary Alphabet ◽  
Self Similar

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a self-similar group of finite type is finite, level-transitive, or topologically finitely generated. Using these criteria and GAP computations we show that for the binary alphabet there is no infinite topologically finitely generated self-similar group given by patterns of depth 3, and there are 32 such groups for depth 4.

Groups acting freely on R-trees

1991 ◽  
Vol 11 (4) ◽  
pp. 737-756 ◽  
Author(s):  
John W. Morgan ◽  
Richard K. Skora
Keyword(s):  
Euler Characteristic ◽  
Cyclic Group ◽  
Abelian Subgroup ◽  
Closed Surface ◽  
Infinite Cyclic Group ◽  
Precise Result ◽  
Finitely Presented Group ◽  
Hnn Extension ◽  
Image Position ◽  
Finitely Presented

AbstractIn this paper we study the question of which groups act freely on R-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case.Let the finitely presented group G act freely on an R-tree. If A is a non-cyclic, abelian subgroup of G, then A is contained in an abelian subgroup A′ which is a free factor of G.The second part of the paper concerns groups whch split as an HNN-extension along an infinite cyclic group. Here is one formulation of our main result in that case.Let the finitely presented group G act freely on an R-tree. If G has an HNN-decompositionwhere (s) is infinite cyclic, then there is a subgroup H′ ⊂ H such that either(a); or(b),where S is a closed surface of non-positive Euler characteristic.A slightly different, more precise result is also given.

AN ITERATED FUNCTION SYSTEM FOR THE CLOSURE OF THE ADDING MACHINE GROUP

Fractals ◽  
2015 ◽  
Vol 23 (03) ◽  
pp. 1550033 ◽  
Author(s):  
MUSTAFA SALTAN ◽  
BÜNYAMİN DEMİR
Keyword(s):  
Automorphism Group ◽  
Iterated Function System ◽  
Rooted Tree ◽  
Function System ◽  
Similar Group ◽  
Machine Group ◽  
Iterated Function ◽  
Self Similar ◽  
The Adding Machine

In this paper, first we equip the automorphism group of the p-ary rooted tree X* with a natural metric and define a family of contractions on Aut(X*). Then, we construct an iterated function system (IFS) whose attractor is the closure of the adding machine group on Aut(X*). Finally, we show that this group is a strong self-similar group in the sense of IFS.

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