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There is the Dog School of Mathematics of dogpile fame. They have a nice tutorial on group theory.
Group theory is a branch of mathematics that deals with the study of groups, which are mathematical structures that consist of a set of elements and a binary operation that combines any two elements to form a third element. It is used to study symmetry and patterns in various fields such as physics, chemistry, and computer science.
The basic concepts of group theory include groups, subgroups, cosets, homomorphisms, and isomorphisms. Groups are sets of elements with a binary operation, subgroups are subsets of groups that also form groups, cosets are subsets of groups that are obtained by multiplying a subgroup by a fixed element, homomorphisms are functions that preserve the group structure, and isomorphisms are bijective homomorphisms.
Group theory has applications in various fields such as physics, chemistry, computer science, and cryptography. In physics, it is used to study symmetries in physical systems and in particle physics. In chemistry, it is used to study molecular structures and chemical reactions. In computer science, it is used in the design and analysis of algorithms and data structures. In cryptography, it is used to design secure encryption algorithms.
There are many resources available for learning group theory, including textbooks, online courses, and video lectures. Some recommended textbooks include "Abstract Algebra" by Dummit and Foote, "A First Course in Abstract Algebra" by Fraleigh, and "Group Theory" by Rotman. Online courses and video lectures can be found on websites such as Coursera, Khan Academy, and YouTube.
Some important theorems in group theory include Lagrange's theorem, which states that the order of a subgroup must divide the order of the group, the first and second isomorphism theorems, which relate the structure of a group to its subgroups and homomorphisms, and the Sylow theorems, which provide information about the number of subgroups of a given order in a finite group.