In abstract mathematics, a very large topic is the concept of a group. This is studied in Group Theory, which at a mathematical level is the study of symmetry in a very abstract way (groups usually manifest themselves in nature in forms of symmetry) [5]. Recently, there have been various breakthroughs in group theory, such as the Classification of Finite Simple Groups (the longest mathematical proof) [7], and the three-hundred page proof that all odd-ordered groups are solvable, which won the Abel prize [6].

A group is a set of objects, called elements, that, when paired with an operation , satisfy three axioms: closure (for all elements a and b in the set, ab is also in the set), associativity (for all three elements a, b, and c, a(bc)=(ab) c), existence of an identity (there exists an element e such that for all elements a, ea=ae=a) and existence of inverses (for all elements a, there exists an element a-1 such that aa-1=e). From these axioms, a few simple consequences arise, and group theory is the study of these consequences [5].

Here is an example of a group (this group is known as the dihedral group). If we take a triangle, we can create a group with three elements. If we set the element e as the element that does nothing to the triangle, e would be the identity. We can then say that α is the element that turns the triangle 120° clockwise and α2 turns the triangle 240° clockwise. This set {e, α, α2} is associative, has closure, has an identity, and has inverses. One thing that should be mentioned, because it will be useful in the future, is the idea of a generator. If we say that e=α0, then we can say that all elements in the group can be represented as a power of α. This means that α is a generator of the group. The more complex groups can have many generators [8].

The last axiom, the existence of inverses, has caused problems in groups, because in some groups the inverse is not immediately obvious. One good example of such a group is the Rubiks...

A group is a set of objects, called elements, that, when paired with an operation , satisfy three axioms: closure (for all elements a and b in the set, ab is also in the set), associativity (for all three elements a, b, and c, a(bc)=(ab) c), existence of an identity (there exists an element e such that for all elements a, ea=ae=a) and existence of inverses (for all elements a, there exists an element a-1 such that aa-1=e). From these axioms, a few simple consequences arise, and group theory is the study of these consequences [5].

Here is an example of a group (this group is known as the dihedral group). If we take a triangle, we can create a group with three elements. If we set the element e as the element that does nothing to the triangle, e would be the identity. We can then say that α is the element that turns the triangle 120° clockwise and α2 turns the triangle 240° clockwise. This set {e, α, α2} is associative, has closure, has an identity, and has inverses. One thing that should be mentioned, because it will be useful in the future, is the idea of a generator. If we say that e=α0, then we can say that all elements in the group can be represented as a power of α. This means that α is a generator of the group. The more complex groups can have many generators [8].

The last axiom, the existence of inverses, has caused problems in groups, because in some groups the inverse is not immediately obvious. One good example of such a group is the Rubiks...

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