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TaylorWatts
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Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?
An isomorphism between cyclic groups is a bijective function that preserves the group structure between two cyclic groups. This means that the elements of one group can be mapped to the elements of the other group in a way that respects the group operations.
No, not all cyclic groups can be isomorphic to each other. Cyclic groups are isomorphic if and only if they have the same order. Therefore, two cyclic groups with different orders cannot be isomorphic to each other.
To prove that two cyclic groups are isomorphic, you need to show that there exists a bijective function between them that preserves the group structure. This can be done by showing that the function is both injective (one-to-one) and surjective (onto) and that it respects the group operations.
No, not all elements of a cyclic group are isomorphic to each other. Isomorphic elements must have the same order, so only elements with the same order can be isomorphic to each other within a cyclic group.
Isomorphisms between cyclic groups allow us to identify and understand the underlying structure and properties of different groups. They also allow us to simplify complex group operations and make comparisons between different groups. Isomorphisms play a crucial role in many areas of mathematics, including group theory, abstract algebra, and number theory.