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We introduce and study two classes: minimal non-sofic groups and $\\omega$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group.\n  On the other hand, we show that locally graded non-sofic groups are necessarily $\\om","authors_text":"K{\\i}van\\c{c} Ersoy","cross_cats":[],"headline":"If a minimal non-sofic group has a finitely generated residually finite maximal normal subgroup, then that subgroup is central and the group is a perfect central extension of a finitely generated non-amenable simple group.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-04-21T07:38:54Z","title":"On minimal non-sofic and $\\omega$-non-sofic groups"},"references":{"count":13,"internal_anchors":0,"resolved_work":13,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"M. Gromov. Endomorphisms of symbolic algebraic varieties.J. Eur. Math. Soc. (JEMS), 1(2):109–197, 1999","work_id":"fc075689-a58e-482f-922d-6a86aad64259","year":1999},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"V. Capraro and M. Lupini.Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture. Lecture Notes in Mathematics 2136. Springer, Cham, 2015","work_id":"f04d4a89-fb76-44ec-bf68-1d9bb431b5de","year":2015},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"B. H. Neumann. Some remarks on infinite groups.J. Lond. Math. Soc., 12:120–127, 1937","work_id":"14fd0205-cb8e-4f0c-8268-5be6e785e574","year":1937},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Brescia, K.ıvanç Ersoy, and M","work_id":"a7457245-cfac-4cb5-9f71-2555102360a2","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"S. V. Ivanov and A. Yu. Ol’shanskii. Some applications of graded diagrams in com- binatorial group theory. InGroups, Vol. 2, Proc. Int. Conf., St. Andrews/UK 1989, London Math. Soc. Lecture Note Ser. 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