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Gear300
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To be sure of things, groups do not necessarily have to have only one operation; they may have more, but there is one and only one operation they are identified with. Am I right?
Gear300 said:Thus, if I were asked whether the set of all rational numbers Q with addition was a group, would it be valid to assume multiplication as an operation and state that a/b + c/d = (ad + cb)/(bd) ε Q (for a, b, c, d ε Z) in the case of proving that addition is closed with respect to Q.
In mathematics, a group is a set of elements that is closed under a single operation, typically denoted by *. This means that any two elements in the group can be combined using the operation to produce a third element, and that third element is also in the group.
Yes, a group can have multiple operations as long as they satisfy certain properties. These properties include closure (the result of the operation is always in the group), associativity (the order in which operations are performed does not matter), identity (there exists an element that when combined with any other element using the operation yields that same element), and inverse (every element has an inverse that when combined using the operation yields the identity element).
An example of a group with more than one operation is the set of real numbers under addition and multiplication. Another example is the set of 2x2 matrices with nonzero determinant under matrix multiplication.
Understanding that groups can have more than one operation helps us to explore more complex structures in mathematics. It also allows us to see connections between seemingly unrelated concepts and to generalize our understanding of groups.
Yes, there are certain restrictions on the operations that can be defined on a group. For example, the operations must be well-defined and satisfy the group properties mentioned earlier. Additionally, the operations must be compatible with each other, meaning that they must interact in a consistent and meaningful way.