A random base change algorithm for permutation groups

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

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IRREDUCIBLE ABELIAN GROUPS OF AUTOMORPHISMS OF AN ABELIAN GROUP, AND PRIMITIVE SOLVABLE PERMUTATION GROUPS

Izvestiya Mathematics ◽

10.1070/im1995v044n02abeh001603 ◽

1995 ◽

Vol 44 (2) ◽

pp. 395-402 ◽

Cited By ~ 1

Author(s):

K K Shchukin

Keyword(s):

Abelian Group ◽

Abelian Groups ◽

Permutation Groups ◽

Groups Of Automorphisms

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Developing a Genetic System in Deinococcus radiodurans for Analyzing Mutations

Genetics ◽

10.1093/genetics/166.2.661 ◽

2004 ◽

Vol 166 (2) ◽

pp. 661-668

Author(s):

Mandy Kim ◽

Erika Wolff ◽

Tiffany Huang ◽

Lilit Garibyan ◽

Ashlee M Earl ◽

...

Keyword(s):

Dna Sequences ◽

Deinococcus Radiodurans ◽

Genetic System ◽

Base Change ◽

Base Substitution ◽

Wild Type ◽

Base Pairs ◽

E Coli ◽

Radiation Induced ◽

Induced Mutagenesis

Abstract We have applied a genetic system for analyzing mutations in Escherichia coli to Deinococcus radiodurans, an extremeophile with an astonishingly high resistance to UV- and ionizing-radiation-induced mutagenesis. Taking advantage of the conservation of the β-subunit of RNA polymerase among most prokaryotes, we derived again in D. radiodurans the rpoB/Rif r system that we developed in E. coli to monitor base substitutions, defining 33 base change substitutions at 22 different base pairs. We sequenced >250 mutations leading to Rif r in D. radiodurans derived spontaneously in wild-type and uvrD (mismatch-repair-deficient) backgrounds and after treatment with N-methyl-N′-nitro-N-nitrosoguanidine (NTG) and 5-azacytidine (5AZ). The specificities of NTG and 5AZ in D. radiodurans are the same as those found for E. coli and other organisms. There are prominent base substitution hotspots in rpoB in both D. radiodurans and E. coli. In several cases these are at different points in each organism, even though the DNA sequences surrounding the hotspots and their corresponding sites are very similar in both D. radiodurans and E. coli. In one case the hotspots occur at the same site in both organisms.

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