Finding if two groups are isomorphic

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In summary, the group {U(7), *} and {Z(6), +} are being compared for isomorphism. The tables for each group were drawn and it was noticed that they are the same size, with the identity elements being 1 for U7 and 0 for Z6. However, it was found that the highest order for U7 is 7, which does not appear as an order in Z6. The question asks to show that the two groups are isomorphic, but the difference in highest orders raises the possibility that they may not be. Further clarification on the definition of U7 is needed.
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sugarplum31
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Homework Statement



Show that the group {U(7), *} is isomorphic to {Z(6), +}

Homework Equations





The Attempt at a Solution



I drew the tables for each one. I can see that they are the same size and the identity element for U7 is 1, and for Z6 is 0. I don't really see any relationship between the two tables. I found the highest order for U7 is 7, which does not appear as an order in Z6. I thought that for two groups to be isomorphic, if one group had an element with an order of X, the other group also had to have a group with that order.

The question makes me think that the two groups ARE isomorphic since it says "show" that they are. Is it possible that they are not? Thanks for hte help!
 
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  • #2
You might want to define what U7 is. Is it the multiplicative subgroup of Z7? If so, then like Z6 it only has six elements. How can it have an element of order 7?? You should probably check your tables.
 

What is the concept of isomorphism in group theory?

Isomorphism in group theory refers to the similarity between two groups that have the same structure and can be mapped onto each other. This means that the two groups have the same number of elements, the same operations, and the same group structure.

How do you determine if two groups are isomorphic?

In order to determine if two groups are isomorphic, you must show that there exists a bijective function (one-to-one and onto) between the two groups that preserves the group operations. This means that the function must map the elements of one group to the corresponding elements of the other group, and the group operations must be the same in both groups.

What are the properties of isomorphic groups?

Isomorphic groups have the same order (or number of elements), the same group operations (such as addition, multiplication, or composition), and the same group structure. This means that the elements of the two groups can be rearranged in the same way without changing the group's properties.

Can two groups have the same elements but not be isomorphic?

Yes, two groups can have the same elements but not be isomorphic. This is because the group operations and structure may differ, even if the elements are the same. For example, the group of integers under addition and the group of integers under multiplication have the same elements, but they are not isomorphic.

How is isomorphism useful in mathematics and science?

Isomorphism is useful in mathematics and science because it allows us to study groups with different structures and operations by finding equivalent groups that are easier to understand. Isomorphic groups have the same properties, so any results or theorems that hold for one group will also hold for the other group.

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