# Every finite abelian p-group is isomorphic to a product of cyclic groups of p-power order in Abstract Algebra: A first course by Dan Sarasino

I am teaching myself abstract algbera with the book Abstract Algebra: A first course by Dan Sarasino.

At section 14, 14.1, the author proves every finite $$p$$-group is isomorphic to a product of cyclic groups of $$p$$-power order

Essential details:

Given a finite abelian $$p$$-group $$G$$ with order $$|G| > 1$$, choose $$x \neq e$$ and let $$A = \langle x \rangle$$ being the cyclic group generated by $$x$$. By the induction hypothesis, i.e any group with order less than $$|G|$$ can be decomposed into a direct product of cyclic groups of $$p$$-power order.

$$G / A \cong \langle y_1 \rangle \times \langle y_2 \rangle \times \ldots \times \langle y_m \rangle$$

where $$y_1, y_2, \ldots, y_m$$ have orders $$p^{t_1}, p^{t_2}, \ldots, p^{t_m}$$ for some integers $$t_1, t_2, \ldots, t_m$$

The part I don't understand

The author claimed that every element in $$G / A$$ has a unique representation

$$(Ax_1)^{r_1} (Ax_2)^{r_2} \ldots (Ax_m)^{r_m}$$

with $$0 \leq r_i < p^{t_i}$$ for $$1 \leq i \leq m$$

Wouldn't the unique representation of element in $$G / A$$ being the $$m$$-tuple of $$y_i^{r_i}$$? How did the author derive the product representation?

$$G / A \cong \langle y_1 \rangle \times \langle y_2 \rangle \times \ldots \times \langle y_m \rangle$$
The author's notation is then tacitly identifying an internal direct product with an external direct product. What this means is that we can choose $$x_i \in G$$, such that under the above isomorphism, $$x_iA \in G/A$$ corresponds to $$(y_1^0, \ldots, y_{i-1}^0, y_i, y_{i+1}^0, \ldots, y_m^0)$$ in $$\langle y_1 \rangle \times \langle y_2 \rangle \times \ldots \times \langle y_m \rangle$$. Then $$(Ax_1)^{r_1} (Ax_2)^{r_2} \ldots (Ax_m)^{r_m}$$ corresponds to $$(y_1^{r_1}, \ldots, y_m^{r_m})$$ and expressions of this form in the $$x_i$$ and $$r_i$$ will give unique representations of each element of $$G/A$$ subject to the given restrictions on the $$r_i$$.