A lattice walk with all steps having the same length $d$ is called a $d$-walk. Denote by ${\mathcal{T}}_{d}$ the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a $d$-walk. We examine some geometric and algebraic properties of the terminal set. After observing that $({\mathcal{T}}_{d},+)$ is a normal subgroup of the group $(\mathbb{Z}^{N},+)$, we ask questions about the quotient group $\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of $\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ in terms of $d$. To establish this result, we use several consequences of Fermat’s theorem about representations of prime numbers of the form $4k+1$ as the sum of two squares. One of the consequences is the fact, observed by Sierpiński, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.

As a special case of fully invariant subgroups, strongly invariant subgroups are introduced and studied for Abelian groups.

For a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of with ramification index e. The orders are given explicitly when G is a p-group of order p or p 2. An example is given when G is not abelian.

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.

We identify a condition, which we refer to as cohesiveness, on a subgroup S of the socle G[p] — {x ∊ G : px = 0} of an abelian p-group G which is necessary for S to be the socle of an isotype subgroup of G. It is shown, when S is countable, that this condition is both necessary and sufficient. A further restriction, definable in terms of the coset valuation on G/S, leads to the notion of S being completely cohesive in G. When S is countable, this latter condition is both necessary and sufficient for S to serve as the socle of a balanced subgroup of G. Also noteworthy is the fact that if H and K are, respectively, balanced and isotype subgroups of G with H[p] = K[p], then K is necessarily balanced in G.