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Determine whether or not the following groups are isomorphic. if two groups g and h are isomorphic, find a bijection f: g h and prove it as an isomorphism. if two groups are not isomorphic, explain why.

User StrikeR
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Final answer:

An isomorphism in abstract algebra relates to the structure of groups. To determine if two groups are isomorphic, we need to find a bijection that preserves the group operation and prove it as an isomorphism. A bijection between two groups will map each element from one group to a unique element of the other group, preserving the group structure.

Step-by-step explanation:

Isomorphism is a concept in abstract algebra that relates to the structure of groups. Two groups are isomorphic if there exists a bijection, or a one-to-one correspondence, between their elements that preserves the group operation. To determine if two groups are isomorphic, we need to find such a bijection and prove that it is an isomorphism.

For example, let's consider the groups (Z, +) and (2Z, +) where Z represents the integers and 2Z represents the even integers. We can define the function f: Z -> 2Z as f(n) = 2n. This function maps every integer in Z to its double in 2Z. This bijection preserves addition, as f(a + b) = 2(a + b) = 2a + 2b = f(a) + f(b). Therefore, f is an isomorphism between (Z, +) and (2Z, +), and we can conclude that these two groups are isomorphic.

On the other hand, if we consider the groups (Z, +) and (Q, +) where Q represents the rational numbers, we cannot find a bijection that preserves addition between them. The rational numbers have more elements than the integers, so we cannot map each integer to a unique rational number. Therefore, these two groups are not isomorphic.

User Stanford
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