= Solution
The group $B$ is a <vector space over a finite field>, namely $\mathbb F_2$, because every element has order at most two. It is infinite-dimensional. The complete first-order theory of infinite-dimensional $\mathbb F_2$-vector spaces says, for each $n$, that there are $n$ linearly independent vectors; the usual elimination argument for vector spaces shows that all infinite-dimensional $\mathbb F_2$-vector spaces are elementarily equivalent.
Every countably infinite model of this theory has dimension $\aleph_0$: finite dimension would make it finite, while uncountable dimension would make its underlying set uncountable. Any two vector spaces over the same field with the same dimension are isomorphic. Therefore $\operatorname{Th}(B,+,0)$ is aleph-zero-categorical, as recorded by the <aleph-zero-categoricity of an infinite-dimensional vector space over a finite field>.
Solved by gpt-5.6-sol high.
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