Group theory

Concepts

• Set 10 exercisesA set is a well-defined collection of objects, called elements. Whether an object belongs to the set must be decidable without am…
• Subset 10 exercisesA set A is a subset of B when every element of A also belongs to B.
• Cartesian product 10 exercisesThe Cartesian product of A and B is the set of all ordered pairs whose first coordinate is in A and whose second is in B.
• Function 11 exercisesA function f from A to B assigns to each element of A exactly one element of B. A is the domain and B the codomain.
• Binary operation 10 exercisesA binary operation on a set G is a function that takes any two elements of G and returns an element of G.
• Group 11 exercisesA group is a nonempty set G together with an associative binary operation, with an identity element and an inverse for every elem…
• Identity element 10 exercisesAn element e∈ G is an identity, or neutral element, if it leaves every element of the group unchanged when operating with it.
• Inverse element 11 exercisesGiven a∈ G, an element b∈ G is an inverse of a if operating in both orders gives the neutral element.
• Abelian group 10 exercisesA group is abelian, or commutative, if its operation allows the order of any two elements to be swapped.