Group order notation?

Question

I'm working with Dummit and Foote's Abstract Algebra text, and I encountered some notation that confused me.

The theorem I saw it in reads as follows. Suppose $\varphi:G \rightarrow H$ is a homomorphism of groups. then ker $ \varphi \unlhd G$ and $G$/ker $\varphi$ $\cong$ $\varphi(G)$.

The corollary to the theorem reads $|G:$ ker $\varphi |=|\varphi(G)|$

What does this latter expression mean? I don't understand how to interpret $|G:$ ker $\varphi |$.

Thanks for your help.

Answer 1

Perhaps you are more familiar with the notation $[G : \ker \phi]$? It means that the index of the subgroup $\ker \phi$ in $G$ is equal to the cardinality of $\phi(G)$, that is, the image of $\phi$.

You can read more on the notion of index of a subgroup here. Dummit & Foote introduce the index of a subgroup at page 90 of Chapter 3.