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Author(s):  
Daniele Corradetti

Abstract Recent papers of Todorov and Dubois-Violette[4] and Krasnov[7] contributed revitalizing the study of the exceptional Jordan algebra h3(O) in its relations with the true Standard Model gauge group GSM. The absence of complex representations of F4 does not allow Aut (h3 (O)) to be a candidate for any Grand Unified Theories, but the group of automorphisms of the complexification of this algebra isisomorphic to the compact form of E6. Following Boyle in [12], it is then easy to show that the gauge group of the minimal left-right symmetric extension of the Standard Model is isomorphic to a proper subgroup of Aut(C⊗h3(O))


Author(s):  
Marius Tărnăuceanu

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio [Formula: see text]. This extends the case [Formula: see text] studied in previous papers [R. Brandl and W. Shi, Finite groups whose element orders are consecutive integers, J. Algebra 143 (1991) 388–400; Y. Feng, Finite groups whose abelian subgroup orders are consecutive integers, J. Math. Res. Exp. 18 (1998) 503–506; W. Shi, Finite groups whose proper subgroup orders are consecutive integers, J. Math. Res. Exp. 14 (1994) 165–166].


Author(s):  
OMER ANGEL ◽  
YINON SPINKA

Abstract Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.


2020 ◽  
pp. 1-44
Author(s):  
STEVEN HURDER ◽  
OLGA LUKINA

A Cantor action is a minimal equicontinuous action of a countably generated group $G$ on a Cantor space $X$ . Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group $G$ , we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees.


2020 ◽  
Vol 30 (2) ◽  
pp. 290-304
Author(s):  
S. Worawiset ◽  
◽  
J. Koppitz ◽  

In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups Gα∪Gβ (α>β) with an injective structure homomorphism, where Gα has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.


2019 ◽  
Vol 12 (07) ◽  
pp. 2050003
Author(s):  
M. Farrokhi D. G. ◽  
M. Rajabian ◽  
A. Erfanian

The relative Cayley graph of a group [Formula: see text] with respect to its proper subgroup [Formula: see text] is a graph whose vertices are elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] for some [Formula: see text], where [Formula: see text] is an inverse-closed subset of [Formula: see text]. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants.


2019 ◽  
Vol 18 (11) ◽  
pp. 1950210
Author(s):  
Bijan Taeri ◽  
Fatemeh Tayanloo-Beyg

In this paper we classify finite non-abelian groups having a unique non-abelian proper subgroup.


2019 ◽  
Vol 29 (04) ◽  
pp. 713-722
Author(s):  
Huaguo Shi ◽  
Zhangjia Han ◽  
Heng Lv ◽  
Longhui Zhang

A finite group [Formula: see text] is called a [Formula: see text]-group if every proper subgroup of [Formula: see text] is either quasi-normal or self-normal in [Formula: see text]. In this paper, the authors classify the non-[Formula: see text]-groups whose proper subgroups are all [Formula: see text]-groups.


Author(s):  
Ekaterina V. Zubei

A finite non-nilpotent group G is called a B-group if every proper subgroup of the quotient group G/Φ(G) is nilpotent. We establish the r-solvability of the group in which some Sylow r-subgroup permutes with the derived subgroups of 2-nilpotent (or 2-closed) B-subgroups of even order and the solvability of the group in which the derived subgroups of 2-closed and 2-nilpotent B-subgroups of even order are permutable.


2019 ◽  
Vol 18 (02) ◽  
pp. 1950037
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Feng Tang

Let [Formula: see text] be a finite group. A proper subgroup [Formula: see text] of [Formula: see text] is said to be weakly monomial if the order of [Formula: see text] satisfies [Formula: see text]. In this paper, we determine all the weakly monomial maximal subgroups of finite simple groups.


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