scholarly journals Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy

2021 ◽  
Vol 3 (1) ◽  
pp. 68-79
Author(s):  
Michel Planat ◽  
David Chester ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Fang Fang ◽  
...  
Keyword(s):  
Finite Group ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Genetic Code ◽  
Finite Groups ◽  
Vital Role ◽  
Recent Model ◽  
Biological Functions ◽  
Kummer Surface ◽  
Octahedral Group

The Kummer surface was constructed in 1864. It corresponds to the desingularization of the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2O the binary octahedral group), we now find that groups G6:=(288,69)≅Z6⋊2O and G7:=(336,118)≅Z7⋊2O can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.

scholarly journals Finite Groups for the Kummer Surface: the Genetic Code and a Quantum Gravity Analogy

2021 ◽  
Author(s):  
Michel Planat --- ◽  
David Chester ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Fang Fang ◽  
...  
Keyword(s):  
Finite Group ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Genetic Code ◽  
Finite Groups ◽  
Vital Role ◽  
Recent Model ◽  
Biological Functions ◽  
Kummer Surface ◽  
Octahedral Group

The Kummer surface was constructed in 1864. It corresponds to the desingularisation of the quotient of a 4-torus by 16 complex double points. Kummer surface is kwown to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2O the binary octahedral group), we now find that groups G6:=(288,69)≅Z6⋊2O and G7:=(336,118)≅Z7⋊2O can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.

scholarly journals Complete Quantum Information in the DNA Genetic Code

2020 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Fang Fang ◽  
Klee Irwin
Keyword(s):  
Amino Acids ◽  
Quantum Information ◽  
Symmetric Group ◽  
Genetic Code ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Threefold Symmetry ◽  
Biological Meaning ◽  
Octahedral Group ◽  
Complete Quantum

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G=\mathbb{Z}_5 \rtimes H$ where $H=\mathbb{Z}_2 . S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G=\mathbb{Z}_5 \rtimes GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the $22$ conjugacy classes of $G$ are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the $22$ characters of $G$ attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

scholarly journals Complete Quantum Information in the DNA Genetic Code

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1993
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Fang Fang ◽  
Klee Irwin
Keyword(s):  
Amino Acids ◽  
Quantum Information ◽  
Symmetric Group ◽  
Genetic Code ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Threefold Symmetry ◽  
Biological Meaning ◽  
Octahedral Group ◽  
Complete Quantum

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature G=Z5⋊H where H=Z2.S4≅2O is isomorphic to the binary octahedral group 2O and S4 is the symmetric group on four letters/bases. The second group has signature G=Z5⋊GL(2,3) and points out a threefold symmetry of base pairings. For those groups, the representations for the 22 conjugacy classes of G are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the 22 characters of G attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

scholarly journals Complete Quantum Information in the DNA Genetic Code

2020 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Fang Fang ◽  
Klee Irwin
Keyword(s):  
Amino Acids ◽  
Quantum Information ◽  
Symmetric Group ◽  
Genetic Code ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Threefold Symmetry ◽  
Biological Meaning ◽  
Octahedral Group ◽  
Complete Quantum

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G=\mathbb{Z}_5 \rtimes H$ where $H=\mathbb{Z}_2 . S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G=\mathbb{Z}_5 \rtimes GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the $22$ conjugacy classes of $G$ are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the $22$ characters of $G$ attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

Sign in / Sign up

Export Citation Format

Share Document