Math Problem Statement
Let G_1 and G_2 be cyclic groups, both of finite order k. And define f : G_1 \rightarrow G_2 by f(a^n) = b^n, where a \in G_1 and b \in G_2, show that this map is surjective
Solution
To show that the map , defined by (where and ), is surjective, we must prove that for every element , there exists an such that .
Step-by-step proof:
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Cyclic nature of and :
- Since and are cyclic groups of finite order , their elements can be written as: where and are generators of and , respectively, and , are their respective identities.
- The order of and is , so and .
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Definition of the map :
- The map is defined as , where is an integer modulo (because of the cyclic nature).
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Surjectivity of :
- To show is surjective, we need to demonstrate that every element in is in the image of . In other words, for any , we need to find an such that .
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Existence of a preimage:
- Since , for any , we choose in such that . This is always possible because ranges over all integers modulo as runs through .
- Thus, for any , we have .
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Conclusion:
- Every element of has a corresponding preimage in under . Therefore, is surjective.
Additional Notes:
- The key property used here is that both and are cyclic groups of the same order , so the exponents and can always match modulo .
- Since is defined in terms of the exponents, and the cyclic groups have the same size, maps all elements of onto without omission.
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Related Questions:
- What additional properties can we prove about ? Is it injective?
- How does the surjectivity depend on the groups and having the same order?
- Can this result be generalized to non-cyclic groups?
- How do the properties of generators and influence the map ?
- What happens if and have different orders?
Tip:
Always verify the properties of the group (cyclicity, order, etc.) before analyzing a map, as these heavily influence the results.
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Math Problem Analysis
Mathematical Concepts
Group Theory
Cyclic Groups
Surjectivity of Functions
Homomorphisms
Formulas
f(a^n) = b^n
Theorems
Surjectivity of Group Homomorphisms
Properties of Cyclic Groups
Suitable Grade Level
Undergraduate Level
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