Geodesic
$P$-stable Abelian groups
Algebra i logika, Tome 52 (2013) no. 5, pp. 606-631.

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$(P,a)$-stable and $(P,s)$-stable Abelian groups are described. It is also proved that every Abelian group is $(P,p)$-stable. In particular, results due to M. A. Rusaleev [Algebra Logika, 50, No. 2, 231–245 (2011)] and T. A. Nurmagambetov [Proc. 11th Conf. Math. Logic, Kazan State Univ., Kazan (1992), p. 106] derive from these.
Keywords: $(P,a)$-stable Abelian group, $(P,s)$-stable Abelian group. 
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E. A. Palyutin. $P$-stable Abelian groups. Algebra i logika, Tome 52 (2013) no. 5, pp. 606-631. http://geodesic.mathdoc.fr/item/AL_2013_52_5_a6/

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