Введение
Теория групп
В математике, в частности в теории групп, аффинные группы по классам вычетов – это определенные группы перестановок, действующие на (целые числа), элементы которых являются биективными аффинными отображениями по классам вычетов. Отображение называется аффинным по классу вычетов, если существует ненулевое целое число *a* такое, что ограничения отображения на классы вычетов (mod *a*) все аффинны. Это означает, что для любого класса вычетов *r* существуют коэффициенты *b* и *c* такие, что ограничение отображения на множество {*x* | *x* ≡ *r* (mod *a*)} задается выражением *bx + c*.
In mathematics, specifically in group theory, residue class wise affine
groups are certain permutation groups acting on
(the integers), whose elements are bijective
residue class wise affine mappings. A mapping is called residue class wise affine
if there is a nonzero integer such that the restrictions of
to the residue classes
(mod ) are all affine. This means that for any
residue class there are coefficients
such that the restriction of the mapping
to the set is given by
Residue class wise affine groups are countable, and they are accessible
to computational investigations. Many of them act multiply transitively on or on subsets thereof. A particularly basic type of residue class wise affine permutations are the
class transpositions: given disjoint residue classes
and , the corresponding class transposition is the permutation
of which interchanges and
for every and which
fixes everything else. Here it is assumed that
and that
The set of all class transpositions of generates
a countable simple group which has the following properties:
It is not finitely generated. Every finite group, every free product of finite groups and every free group of finite rank embeds into it. The class of its subgroups is closed under taking direct products, under taking wreath products with finite groups, and under taking restricted wreath products with the infinite cyclic group. It has finitely generated subgroups which do not have finite presentations. It has finitely generated subgroups with algorithmically unsolvable membership problem. It has an uncountable series of simple subgroups which is parametrized by the sets of odd primes. It is straightforward to generalize the notion of a residue class wise affine group
to groups acting on suitable rings other than ,
though only little work in this direction has been done so far. See also the Collatz conjecture, which is an assertion about a surjective,
but not injective residue class wise affine mapping.
Аффинные группы по классам вычетов счетны и доступны для вычислительных исследований. Многие из них действуют мультитранзитивно на ℤ или на подмножествах ℤ. Особенно базовым типом перестановок, аффинных по классу вычетов, являются перестановки классов: для данных непересекающихся классов вычетов *A* и *B*, соответствующая перестановка классов – это перестановка ℤ, которая меняет местами элементы из *A* и *B* для каждого *x* и фиксирует все остальные элементы. Здесь предполагается, что *A* и *B* непусты. Множество всех перестановок классов порождает счетную простую группу, которая обладает следующими свойствами:
In mathematics, specifically in group theory, residue class wise affine
groups are certain permutation groups acting on
(the integers), whose elements are bijective
residue class wise affine mappings. A mapping is called residue class wise affine
if there is a nonzero integer such that the restrictions of
to the residue classes
(mod ) are all affine. This means that for any
residue class there are coefficients
such that the restriction of the mapping
to the set is given by
Residue class wise affine groups are countable, and they are accessible
to computational investigations. Many of them act multiply transitively on or on subsets thereof. A particularly basic type of residue class wise affine permutations are the
class transpositions: given disjoint residue classes
and , the corresponding class transposition is the permutation
of which interchanges and
for every and which
fixes everything else. Here it is assumed that
and that
The set of all class transpositions of generates
a countable simple group which has the following properties:
It is not finitely generated. Every finite group, every free product of finite groups and every free group of finite rank embeds into it. The class of its subgroups is closed under taking direct products, under taking wreath products with finite groups, and under taking restricted wreath products with the infinite cyclic group. It has finitely generated subgroups which do not have finite presentations. It has finitely generated subgroups with algorithmically unsolvable membership problem. It has an uncountable series of simple subgroups which is parametrized by the sets of odd primes. It is straightforward to generalize the notion of a residue class wise affine group
to groups acting on suitable rings other than ,
though only little work in this direction has been done so far. See also the Collatz conjecture, which is an assertion about a surjective,
but not injective residue class wise affine mapping.
Она не конечно порождена. Каждая конечная группа, каждое свободное произведение конечных групп и каждая свободная группа конечного ранга вкладываются в нее. Класс ее подгрупп замкнут относительно взятия прямых произведений, взятия венковых произведений с конечными группами и взятия ограниченных венковых произведений с бесконечной циклической группой. Она имеет конечно порожденные подгруппы, которые не имеют конечных представлений. Она имеет конечно порожденные подгруппы с алгоритмически неразрешимой задачей о принадлежности. Она имеет несчетный ряд простых подгрупп, который параметризуется множествами нечетных простых чисел. Прямолинейно обобщить понятие аффинной группы по классу вычетов на группы, действующие на подходящих кольцах, отличных от ℤ, хотя пока в этом направлении было сделано немного работы. См. также гипотезу Коллатца, которая является утверждением о сюръективном, но не инъективном аффинном отображении по классу вычетов.
In mathematics, specifically in group theory, residue class wise affine
groups are certain permutation groups acting on
(the integers), whose elements are bijective
residue class wise affine mappings. A mapping is called residue class wise affine
if there is a nonzero integer such that the restrictions of
to the residue classes
(mod ) are all affine. This means that for any
residue class there are coefficients
such that the restriction of the mapping
to the set is given by
Residue class wise affine groups are countable, and they are accessible
to computational investigations. Many of them act multiply transitively on or on subsets thereof. A particularly basic type of residue class wise affine permutations are the
class transpositions: given disjoint residue classes
and , the corresponding class transposition is the permutation
of which interchanges and
for every and which
fixes everything else. Here it is assumed that
and that
The set of all class transpositions of generates
a countable simple group which has the following properties:
It is not finitely generated. Every finite group, every free product of finite groups and every free group of finite rank embeds into it. The class of its subgroups is closed under taking direct products, under taking wreath products with finite groups, and under taking restricted wreath products with the infinite cyclic group. It has finitely generated subgroups which do not have finite presentations. It has finitely generated subgroups with algorithmically unsolvable membership problem. It has an uncountable series of simple subgroups which is parametrized by the sets of odd primes. It is straightforward to generalize the notion of a residue class wise affine group
to groups acting on suitable rings other than ,
though only little work in this direction has been done so far. See also the Collatz conjecture, which is an assertion about a surjective,
but not injective residue class wise affine mapping.
Ссылки и внешние ссылки
Штефан Коль. Остаточные классы и аффинные группы. Диссертация, Университет Штутгарта, 2005. Архивный сервер Немецкой национальной библиотеки OPUS Datenbank (Университет Штутгарта)
Штефан Коль. RCWA – Аффинные группы, заданные по остаточным классам. Пакет GAP. 2005. Штефан Коль. Простая группа, порожденная инволюциями, переставляющими остаточные классы целых чисел. Math. Z. 264 (2010), No. 4, 927–938.
Stefan Kohl. RCWA – Residue Class Wise Affine Groups. GAP package. 2005. Stefan Kohl. A Simple Group Generated by Involutions Interchanging Residue Classes of the Integers. Math. Z. 264 (2010), no. 4, 927–938.