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In group theory, what is the direct product of a symmetry group with itself? Say T*T or O*O?
The direct product of a symmetry group with itself is a mathematical operation that combines two symmetry groups to form a new group. It is denoted by the symbol "×" and is a way of expressing the fact that the elements of the two groups can be combined in a specific way to create a new group.
The direct product of a symmetry group with itself is calculated by taking the Cartesian product of the two groups. This means multiplying each element in the first group with each element in the second group. The resulting elements form the new group.
The direct product of a symmetry group with itself is significant because it allows us to understand the symmetries of more complex objects by breaking them down into simpler components. It also helps in studying the relationships between different symmetry groups and their properties.
No, the direct product of a symmetry group with itself is not commutative. This means that the order in which the groups are multiplied matters. In other words, the direct product of group A with group B is not the same as the direct product of group B with group A.
The direct product of a symmetry group with itself is related to group isomorphism in that it can be used to determine whether two groups are isomorphic to each other. If the direct product of two groups is isomorphic to a third group, then the two original groups are also isomorphic to each other.