Geodesic
Residually Finite Algorithmically Finite Groups, Their Subgroups and Direct Products
Matematičeskie zametki, Tome 98 (2015) no. 3, pp. 372-377.

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We construct a finitely generated infinite recursively presented residually finite algorithmically finite group $G$, thus answering a question of Myasnikov and Osin. The group $G$ here is “strongly infinite” and “strongly algorithmically finite”, which means that $G$ contains an infinite Abelian normal subgroup and all finite Cartesian powers of $G$ are algorithmically finite (i.e., for any $n$, there is no algorithm writing out infinitely many pairwise distinct elements of the group $G^n$). We also formulate several open questions concerning this topic.
Keywords: finitely generated group, residually finite group, algorithmically finite group. 
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A. A. Klyachko; A. K. Mongush. Residually Finite Algorithmically Finite Groups, Their Subgroups and Direct Products. Matematičeskie zametki, Tome 98 (2015) no. 3, pp. 372-377. http://geodesic.mathdoc.fr/item/MZM_2015_98_3_a4/

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