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Math and science::Algebra::Aluffi

Group homomorphisms and order

Group homomorphisms and order

Let \( \varphi : G \to H \) be a group homomorphism, and let \( g \in G \) be an element of finite order. Then \( |\varphi(g)| \) divides \( |g| \).


Consequences

  • There are no non-trivial homomorphisms from \( \mathbb{Z}/n\mathbb{Z} \) to \( \mathbb{Z} \), as elements of \( \mathbb{Z}/n\mathbb{Z} \) would need to map to elements of finite order in \( \mathbb{Z} \), of which there is only the identity, \( 0 \).
  • There are no non-trivial homomorphisms from \( C_4 \) to \( C_7 \), as elements of \( C_7 \) other than the identity all have order 7, which doesn't divide 2 or 4 (the orders of non-identity elements of \( C_4 \)). 
  • In general, for a non-trivial group morphism \( C_n \to C_m \) to exist, \( \operatorname{gcd}(n, m) \) must be greater than 1 as elements of \( C_m \) must have an order that divides both \( |C_m| \) and \( |C_n| \).

Source

Aluffi, p66