Additive group of p-adic integers

Definition
Let $$p$$ be a (fixed here) prime number. This is a group determined uniquely up to isomorphism based on $$p$$ and is sometimes denoted $$\mathbb{Z}_{(p)}$$ (though that notation is also used for other things, and we can best infer meaning from context).

As an inverse limit
The additive group of p-adic integers is a profinite group defined as the inverse limit of the inverse system:

$$0 \leftarrow \mathbb{Z}/p\mathbb{Z} \leftarrow \mathbb{Z}/p^2\mathbb{Z} \leftarrow \dots \leftarrow \mathbb{Z}/p^n\mathbb{Z} \leftarrow \dots$$

where each of the maps:

$$\mathbb{Z}/p^{n-1}\mathbb{Z} \leftarrow \mathbb{Z}/p^n\mathbb{Z}$$

reduces an integer mod $$p^n$$ to its value mod $$p^{n-1}$$.

Note that this definition also endows the group with a topology as a profinite group. In this topology, two elements are close if they agree mod $$p^n$$ for large $$n$$.

As sequences with cumulative information
The additive group of p-adic integers is the set of sequences:

$$(a_0, a_1, a_2, \dots, a_n, \dots)$$

where $$a_i$$ is an integer mod $$p^{i+1}$$, the addition is coordinate-wise (with each coordinate addition in the integers mod $$p^{i+1}$$), and for $$i < j$$, reducing $$a_i$$ mod $$p^{j+1}$$ yields $$a_j$$.

As sequences with carries
The additive group of p-adic integers is the set of formal sums:

$$\sum_{i=0}^\infty x_ip^i$$

where $$x_i \in \{ 0,1,2,\dots,p-1 \}$$, and the addition is done with carries, i.e., to add two sequences $$\sum_{i=0}^\infty x_ip^i$$ and $$\sum_{i=0}^\infty y_ip^i$$, we add coordinate-wise and if any of the sums is $$p$$ or more, we take a carry of 1 to the next sum.

As the additive group of the ring of Witt vectors
This group is the additive group of the ring of Witt vectors over the prime field $$\mathbb{F}_p$$.

Powering
The additive group of $$p$$-adic integers is powered over the set of all primes other than $$p$$.

Topological group properties
Here, the topology is from the profinite group structure.

Combining all primes
The profinite completion of the integers, denoted $$\hat {\mathbb{Z}}$$ is a group obtained by taking the profinite completion of the group of integers, which is a residually finite group. Intuitively, this group is obtained by "completing" the group of integers simultaneously at all primes. It turns out that the profinite completion of the integers can be viewed as the external direct product, over all primes $$p$$, of the additive group of $$p$$-adic integers.

Dual concept
The additive group of p-adic integers can, in a vague sense, be considered to be constructed using a method dual to the method used to the quasicyclic group. While the $$p$$-adics are constructed as an inverse limit for surjective maps $$\mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}/p^{n-1}\mathbb{Z}$$, the quasicyclic group is constructed as a direct limit for injective maps $$\mathbb{Z}/p^{n-1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$$.