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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.
A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order. It is known that a CLT group must be solvable and that every supersolvable group is a CLT group. However, there exist solvable groups that are not CLT (for example, A4) and CLT ...
The group consisting of all permutations of a set M is the symmetric group of M. p-group. If p is a prime number, then a p -group is one in which the order of every element is a power of p. A finite group is a p -group if and only if the order of the group is a power of p. p-subgroup. A subgroup that is also a p-group.
Rank of a group. In the mathematical subject of group theory, the rank of a group G, denoted rank (G), can refer to the smallest cardinality of a generating set for G, that is. If G is a finitely generated group, then the rank of G is a non-negative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of ...
Word problem for groups. In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group is the algorithmic problem of deciding whether two words in the generators represent the same element of . The word problem is a well-known example of an undecidable problem.
In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G, [1] or even in every group. [2] Words play an important role in the theory of ...
For a prime p, the p-core of a finite group is defined to be its largest normal p-subgroup.It is the normal core of every Sylow p-subgroup of the group. The p-core of G is often denoted (), and in particular appears in one of the definitions of the Fitting subgroup of a finite group.