MathematicalGroup

A mathematical group is an algebraic entity containing a set of objects and a mathematical operation on those objects. There is a set of rules the objects and operation must obey. Here, A, B, and C, and I are objects in the group, and @ is the group operation.

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The rules:

• <B>Closure</B> If A and B are in the group, A @ B is also in the group.
• <B>Identity</B> There is a special object in the group called I with the properties A @ I = A and also I @ A = A whenever A is in the group.
• <B>Inverse</B> Whenever A is in the group, there is a corrensponding element called the inverse of A also in the group. If A @ B = I and also B @ A = I, then we say B is the inverse of A.
• <B>Associativity</B> ( A @ B ) @ C = A @ ( B @ C ) whenever A, B, C are in the group.

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Examples of mathematical groups:

• integers under addition
• real numbers under addition
• real numbers except zero under multiplication
• integers modulo a prime integer under addition
• a group of rotations in three dimensions
• a Lie group
• one of the sporadic finite simple groups

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As with other abstract algebraic entities, the point of mathematical groups is that you attempt to prove things about groups assuming only the basic properties of mathematical groups given in the definition. That way whatever you prove is useful for all mathematical groups, not just the one you are thinking about at the moment.

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There are other abstract algebraic entities that are groups, but also have additional properties. Some of the more commonly encountered ones are:

• Abelian groups
• Mathematical rings
• Mathematical fields----

You are repeating pages that already exist: see MathematicalGrouP.

The operation has to be binary, You have to prove the uniqueness of the identity and inverse. Our math pages are usually very precise...Let's keep it that way. RoseParks