Showing that groups are isomorphic

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SUMMARY

To demonstrate that two groups are isomorphic, it is sufficient to find a single isomorphism between them. In the case of an infinite cyclic group G with generator g, G is isomorphic to the group of integers Z. The function f defined by f(g) = ord(g) is both bijective and a homomorphism, confirming the isomorphism. This conclusion is established through the properties of the function f, which must be explicitly shown to be bijective and a homomorphism for clarity.

PREREQUISITES
  • Understanding of group theory concepts, particularly isomorphisms.
  • Familiarity with cyclic groups and their generators.
  • Knowledge of bijective functions and homomorphisms.
  • Basic comprehension of the group of integers Z.
NEXT STEPS
  • Study the properties of isomorphisms in group theory.
  • Learn about cyclic groups and their structure in detail.
  • Explore examples of bijective functions and their role in group homomorphisms.
  • Investigate the implications of group isomorphism in abstract algebra.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in understanding the fundamentals of group theory and isomorphisms.

gottfried
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If one wants to show that two groups are isomorphic is simply finding a single isomorphism between them sufficient?

For example.

If G is an infinite cyclic group with generator g show that G is isomorphic to [itex]Z[/itex].

So suppose f(g)=ord(g)

then f is bijective and a homomorphism I believe?
 
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gottfried said:
If one wants to show that two groups are isomorphic is simply finding a single isomorphism between them sufficient?

Yes.

For example.

If G is an infinite cyclic group with generator g show that G is isomorphic to [itex]Z[/itex].

So suppose f(g)=ord(g)

then f is bijective and a homomorphism I believe?

Yes, this is true. However, you might want to give a bit of explanation on why it is bijective and a homomorphism.
 
Cool thanks for clearing it up.
 

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