In Hungerford's Abstract Algebra: An Introduction, he writes The Fundamental Theorem of Finite Abelian Groups as:
"Every finite abelian group G is the direct sum of cyclic groups, each of prime power order.
The following is the proof for this theorem:
My problem is that I do not understand the inductive proof - where is the inductive step? The basis step is that the assertion is true when $H$ has order $2$, which was shown previously in the book, but it looks like they just take the induction step for granted.