Zm × Zn is isomorphic to Zmn iff m and n are coprime
Dependencies:
- Isomorphism on Groups
- Cyclicness is invariant under isomorphism
- Order of element in external direct product
- Order of elements in cyclic group
- Cyclic groups are isomorphic to Z or Zn
- GCD times LCM equals product
Proof
Proof of 'only if' part
Let
Let
Therefore,
Proof of 'if' part
Let
Let
Dependency for: None
Info:
- Depth: 9
- Number of transitive dependencies: 32
Transitive dependencies:
- Group
- Homomorphism on groups
- Mapping of power is power of mapping
- Isomorphism on Groups
- Cyclicness is invariant under isomorphism
- Identity of a group is unique
- Subgroup
- External direct product is a group
- Order of element in external direct product
- Inverse of a group element is unique
- Conditions for a subset to be a subgroup
- Cyclic Group
- Cyclic groups are isomorphic to Z or Zn
- Every number has a prime factorization
- Coprime
- Integer Division Theorem
- GCD is the smallest Linear Combination
- Common divisor divides GCD
- gcd(a1/d, a2/d, ..., an/d) = gcd(a1, a_2, ..., an)/d
- If x divides ab and x is coprime to a, then x divides b
- Order of elements in cyclic group
- Euclid's lemma
- Fundamental Theorem of Arithmetic
- Prime Factorization Exponent List (PFEL)
- PFEL of ratio is difference of PFEL
- PFEL of product is sum of PFEL
- Divisible iff PFEL is less than or equal
- PFEL of lcm is max of PFEL
- PFEL of gcd is min of PFEL
- LCM divides common multiple
- Product of coprime divisors is divisor
- GCD times LCM equals product