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greatscott
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When physicists say "elementary particles form a group," what kind of operations and sets are in question? (I presume, a group consists of a set and an operation)
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A group is a mathematical concept that consists of a set of elements and an operation that combines any two elements in the set to produce another element in the set.
The operation in a group is used to define the relationship between the elements in the set. It allows us to perform calculations and determine the properties of the group.
There are four main properties that a group must satisfy: closure, associativity, identity, and invertibility. Closure means that the operation must produce an element within the group. Associativity means that the order in which the operation is performed does not matter. Identity means that there is an element in the group that when combined with any other element in the group, produces that same element. Invertibility means that every element in the group has an inverse that when combined with the element produces the identity element.
Yes, a group can have multiple operations as long as all the operations satisfy the four properties mentioned above. This is known as a group with multiple operations or a multi-operation group.
Groups are used in various branches of science, including mathematics, physics, chemistry, and computer science. They provide a way to study and understand the structure and behavior of different systems. For example, groups are used to describe the symmetries of molecules in chemistry and the rotational and translational motions of particles in physics.