Yes, it should be |G'|. Thank you for catching that!

[ehrenfest]

[SOLVED] group theory

Homework Statement

Let $\phi:G \to G'$ be a group homomorphism. Show that if |G| is finite, then $|\phi(G)|$ is finite and is a divisor of |G|.

Homework Equations

The Attempt at a Solution

Should the last word be |G'|? Then it would follow from Lagrange's Theorem.

 

[Hurkyl]

Nope; it's right as stated. (And can also use Lagrange's theorem in its proof)

 

[quasar987]

Think first isomorphism theorem.

 

[ehrenfest]

I haven't gotten to the first isomorphism theorem yet, but I don't even need it:

We know that $\phi^{-1}(\phi(a)) = aH = Ha$, where H = Ker(phi). So, the cardinality of phi(G) will be the index of H in G, which must divide |G| by Lagrange's Theorem.

Is that right?

 

[Mathdope]

Bingo.