Set, binary operation, group, abelian group, subgroup, cyclic group, permutation, normal subgroup, homomorphism, isomorphism
A collection of distinct objects. In group theory, these objects are often numbers, functions, or other mathematical entities.
Group: A set with a binary operation that is closed, associative, has an identity element, and an inverse element for every element in the set. Groups are one of the most fundamental algebraic structures.
Abelian group: A group where the binary operation is commutative. This means the order in which two elements are combined does not affect the result. The integers under addition form an abelian group.
Subgroup: A subset of a group that is also a group under the same binary operation.
Cyclic group: A group that can be generated by a single element. This means every element in the group can be expressed as a power of that single element.
Permutation: A rearrangement of the elements of a set. The set of all permutations of \(n\) elements forms a group called the symmetric group, \(S_{n}\).
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