In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after Niels Henrik Abel.
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements. Some of these variants are still open questions.
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing by any number field.
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.
A group is a set together with an associative operation that admits an identity element and such that there exists an inverse for every element.
In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n.
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.
In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.
In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module consisting entirely of torsion elements. A module is torsion-free if its only torsion element is the zero element.
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group.
In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite.
In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem.
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate growth. The group was originally constructed by Grigorchuk in a 1980 paper and he then proved in a 1984 paper that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968. The Grigorchuk group remains a key object of study in geometric group theory, particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of iterated monodromy groups.
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem.
In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.
Evgenii Solomonovich Golod was a Russian mathematician who proved the Golod–Shafarevich theorem on class field towers. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem.
In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first proved by Igor Shafarevich, though Alexander Schmidt later pointed out a gap in the proof, which was fixed by Shafarevich.
