May 1, 2024
3 minute read
Group theory is the study of groups, which are algebraic structures that are characterized by an operation called multiplication. Groups are used in a wide variety of mathematical applications, including algebra, geometry, number theory, and topology. They also have applications in physics, computer science, and other fields.
Origins
The origins of group theory can be traced back to the work of Évariste Galois in the 19th century. Galois was interested in finding a way to solve polynomial equations. He developed a theory of groups that allowed him to determine whether or not a given polynomial equation was solvable. Galois's work was later extended by other mathematicians, and group theory has become a major branch of mathematics.
Definitions
A group is a set of elements together with an operation that combines any two elements of the set to form a third element of the set. The operation is usually denoted by a symbol such as addition (+), multiplication (×), or juxtaposition. The following are some of the most important properties of groups:
- The operation is associative, meaning that for any elements a, b, and c in the group, we have (a × b) × c = a × (b × c).
- There is an identity element, which is an element that, when combined with any other element in the group, leaves that element unchanged. The identity element is usually denoted by the symbol 1 or e.
- For each element a in the group, there is an inverse element, which is an element that, when combined with a, gives the identity element. The inverse element of a is usually denoted by the symbol a-1.
Types of Groups
There are many different types of groups. Some of the most common types of groups include:
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Find a path to becoming a Group Theory. Learn more at:
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Reading list
We've selected 13 books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Group Theory.
Provides a comprehensive introduction to group theory, including both finite and infinite groups. It is suitable for both undergraduate and graduate students.
Provides a comprehensive treatment of group theory, including character theory. It is suitable for both undergraduate and graduate students.
Provides a comprehensive treatment of the representation theory of finite groups, which is an important tool in many areas of mathematics. It is suitable for graduate students.
Provides a comprehensive treatment of the classification of finite simple groups, which major achievement in mathematics. It is suitable for graduate students.
Provides a comprehensive introduction to abstract algebra, including a thorough treatment of group theory. It is suitable for both undergraduate and graduate students.
Provides a geometric approach to group theory, focusing on the symmetries of geometric objects. It is suitable for both undergraduate and graduate students.
Provides a comprehensive treatment of permutation groups, including both finite and infinite groups. It is suitable for both undergraduate and graduate students.
Provides a comprehensive treatment of the symmetric group, which is an important group in many areas of mathematics. It is suitable for both undergraduate and graduate students.
Provides an introduction to Lie groups and Lie algebras, which are important in many areas of mathematics and physics. It is suitable for both undergraduate and graduate students.
Provides a clear and concise introduction to abstract algebra, including a chapter on group theory. It is suitable for undergraduate students.
Provides a computational approach to group theory, using computer algebra systems. It is suitable for both undergraduate and graduate students.
Provides an introduction to the relationship between group theory and topological manifolds. It is suitable for both undergraduate and graduate students.
Provides an introduction to algebraic geometry, which branch of mathematics that uses group theory to study geometric objects. It is suitable for both undergraduate and graduate students.
For more information about how these books relate to this course, visit:
OpenCourser.com/topic/tv2xoo/group
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