Solution to the sandwich classification problem in arbitrary groups and applications to classical-like groups over arbitrary rings

Abstract: Let G be an arbitrary group and F an arbitrary subgroup. For each mixed commutator subgroup K = [F, H] of G, we define the notion of an F-cocommutator subgroup over K. The set of F-cocommutator subgroups over K forms a sandwich of subgroups of G, which is denoted by Sand(K). It has a largest member C(K) called the full cocommutator subgroup over K and if F is perfect then K is its smallest member. C(K) is the replacement in the setting of arbitrary groups for the notion of full congruence subgroup in the setting of classical-like groups over rings when F is the elementary subgroup and the K's are replacements for the relative elementary subgroups of a classical-like group. The MAIN THEOREM is: A subgroup H of G is F-normal if and only if it belongs to a sandwich Sand(K) for some K. Moreover K is unique. We show that the known classification of E-normal subgroups of a classical-like group G(R) over a quasi-finite ring R, where E is the elementary subgroup of G(R), is a consequence of the Main Theorem and we use the Main Theorem to extend this result to classical-like groups G(R) over an arbitrary ring R.

operator algebrasrings and algebras

Audience: researchers in the topic

Western Sydney University Abend Seminars

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