Finding if two groups are isomorphic

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SUMMARY

The groups {U(7), *} and {Z(6), +} are not isomorphic. The identity element for U(7) is 1, while for Z(6) it is 0. The highest order of elements in U(7) is 7, which does not exist in Z(6), where the maximum order is 6. Therefore, the requirement for isomorphism, which states that corresponding elements must have the same order, is not satisfied.

PREREQUISITES
  • Understanding of group theory concepts, specifically isomorphism.
  • Familiarity with the structure of the groups U(7) and Z(6).
  • Knowledge of group orders and identity elements.
  • Ability to construct and analyze Cayley tables for groups.
NEXT STEPS
  • Study the properties of cyclic groups and their orders.
  • Learn about the structure and properties of the multiplicative group U(n).
  • Investigate the criteria for group isomorphism in detail.
  • Explore examples of isomorphic and non-isomorphic groups for practical understanding.
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Students of abstract algebra, mathematicians studying group theory, and anyone interested in understanding group isomorphism and its implications.

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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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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.
 

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