Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

Question

Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$

Can anyone provide references to places where this (or related) groups arise in the literature? The groups $V_4$ and $V_8$ have recently cropped up in a problem I am working on.

The product is often used in defining the $a$-adic integers (the $a$-adic integers are a subring/subgroup of the product.) — Thomas Andrews, Feb 02 '16 at 00:28
@ThomasAndrews the notation here is the algebraic direct sum not the product. Does that not usually mean that all but finitely many entries of any element are $0$? — Mathmo123, Feb 02 '16 at 01:00
I realize that. @Mathmo123 The question asked for "or related groups." — Thomas Andrews, Feb 02 '16 at 01:01

score 2 · Accepted Answer · answered Feb 02 '16 at 03:44

As far as I know, this group does not naturally appear anywhere.

A closely related group is the "Prufer $a$-group" (in quotes because it is usually only defined for $a$ a prime), which is a certain quotient of this group. Its Pontryagin dual is the "$a$-adic integers" (again, in quotes because it is usually only defined for $a$ a prime), which is a certain subgroup of the corresponding infinite product. Both of these groups show up in various places.

Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

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