Finite Abelian Groups (see, for example, [@Herstein99a Sec. 10]) we obtain the following lemma. In this context, a [*partition*]{} of a positive integer $m$ is a finite sequence $(t_i)_{i=1}^r$, $r\in {\ensuremath{\mathbb{N}}}$, of positive integers $t_i$ with the two properties (i) $t_1 \ge t_2 \ge  \dots \ge t_r$, and (ii) $t_1+t_2+ \dots + t_r = m$.

\[ch:intro:cor:FTFAG\] The non-isomorphic groups of order $2^m$, $m\in {\ensuremath{\mathbb{N}}}$, are given by the product groups $$({\ensuremath{\mathbb{Z}}}/2^{t_1}{\ensuremath{\mathbb{Z}}}) \times ({\ensuremath{\mathbb{Z}}}/2^{t_2}{\ensuremath{\mathbb{Z}}}) \times \dots \times ({\ensuremath{\mathbb{Z}}}/2^{t_r}{\ensuremath{\mathbb{Z}}}),$$ where $(t_i)_{i=1}^r$ is a partition of $m$.

Hence, in view of Lemma \[ch:intro:cor:FTFAG\], an addition on the set $\mathcal{A}_2^m$ is defined if we choose a partition $m=t_1+ t_2 +\dots + t_r$ of $m$ and put $$\label{eqn:2-adic}
(\mathcal{A}_2^m, +) \cong ({\ensuremath{\mathbb{Z}}}/2^{t_1}{\ensuremath{\mathbb{Z}}}) \times ({\ensuremath{\mathbb{Z}}}/2^{t_2}{\ensuremath{\mathbb{Z}}}) \times \dots \times ({\ensuremath{\mathbb{Z}}}/2^{t_r}{\ensuremath{\mathbb{Z}}}).$$ Here, the symbol “$\cong$” denotes that the two groups are isomorphic. From the structure of the factors in (\[eqn:2-adic\]) we obtain the following information.

\[cor:types\] The only two types of binary operations on (sub)vectors of digits that may appear in the group law of the abelian group $(\mathcal{A}_2^m, +)$ are the following:

-   addition given by finite product groups of the form $({\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})\times \dots \times ({\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$, which is what we have called addition without carry, or

-   