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Plus teacher and student package: Group Theory This package brings together all the articles on group theory from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of 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.
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.
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.
The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics (for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory and the theory of topological manifolds.
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups.. The isomorphism problem was formulated by Max Dehn, [1] and together with the word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911. [2]
In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems.As in other areas of mathematics, such problems are often made public at professional conferences and meetings.
This is the first known example of an SQ-universal group. Many more examples are now known: Adding two generators and one arbitrary relator to a nontrivial torsion-free group, always results in an SQ-universal group. [2] Any non-elementary group that is hyperbolic with respect to a collection of proper subgroups is SQ-universal. [3]