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goody
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here f:Z36 to Z4 x Z9 is group isomorphism given by f(n+36Z)= (n+4Z, n+9Z) then what is the inverse of f ?
goody said:here f:Z36 to Z4 x Z9 is group isomorphism given by f(n+36Z)= (n+4Z, n+9Z) then what is the inverse of f ?
A group isomorphism is a type of mathematical function that preserves the structure and relationships between elements in a group. It is a one-to-one and onto mapping between two groups that maintains the group operation and identity element.
In this problem, the group isomorphism f is defined as f(n+36Z)= (n+4Z, where n is an integer. This means that every element in the group Z36 is mapped to an element in the group Z4 x Z9, and the operation of addition is preserved.
The notation "n+36Z" represents a coset in the group Z36, where n is an integer and 36Z represents the subgroup of multiples of 36. This notation is used to show that f is a function from the group Z36 to Z4 x Z9.
The group operation in Z36 is addition, so for any two elements a and b in Z36, f(a+b+36Z) = f(a+36Z) + f(b+36Z). This means that the sum of two elements in Z36 is equal to the sum of their corresponding elements in Z4 x Z9 under the group isomorphism f.
Yes, f is both a group isomorphism and a homomorphism. This means that it preserves the group operation and the group structure. In other words, f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b) for all elements a and b in Z36.