Some subgroups defined by identities

On a Problem of P. Hall for Engel Words Ii

“…We recall some notions from [12,13,19,21] to formulate our last main theorem. The first partial margin of E 3 (G) is the set A(G) = {a ∈ G | [x, y, y, y] = [ax, y, y, y] ∀x, y ∈ G} and the first partial margin of E 4 (G) is the set B(G) = {a ∈ G | [x, y, y, y, y] = [ax, y, y, y, y] ∀x, y ∈ G}.…”

“…The first partial margin of E 3 (G) is the set A(G) = {a ∈ G | [x, y, y, y] = [ax, y, y, y] ∀x, y ∈ G} and the first partial margin of E 4 (G) is the set B(G) = {a ∈ G | [x, y, y, y, y] = [ax, y, y, y, y] ∀x, y ∈ G}. Both are characteristic subgroups of G and their properties are described in [1,3,12,13,19,21]. Theorem 1.3.…”

“…Proof of Theorem 1.2. From [2,[11][12][13], we know that E * 2 (G) and E * 3 (G) are characteristic subgroups of G. Of course, this is true also for Z(G), Z 2 (G), Z 3 (G) and Fit(G). From Lemma 2.7, T 3 (G) is a subgroup of G and it is fixed under the action of inner automorphisms of G by results in [11][12][13].…”

“…G) = [x, y, y, y] | ∀x, y ∈ G , E 4 (G) = [x, y, y, y, y] | ∀x, y ∈ G ,E * 3 (G) = {a ∈ G | [x, y, y, y] = [ax, y, y, y] = [x, ay, ay, ay] ∀x, y ∈ G}, E * 4 (G) = {a ∈ G | [x, y, y, y, y] = [ax, y, y, y, y] = [x,ay, ay, ay, ay] ∀x, y ∈ G}. These are always characteristic subgroups of G (see[12][13][14]21]) and dual in the sense of [19, Theorems 1.1 and 1.2]. We also use the setsR 3 (G) = {x ∈ G | [x, y, y, y] = 1 ∀y ∈ G}, R 4 (G) = {x ∈ G | [x, y, y, y, y] = 1 ∀y ∈ G}, L 3 (G) = {x ∈ G | [y, x, x, x] = 1 ∀y ∈ G}, L 4 (G) = {x ∈ G | [y, x, x, x, x] = 1 ∀y ∈ G},It is my pleasure to thank both IMPA and UFRJ for the project CAPES with ref.…”

Some remarks on aperiodic elements in locally nilpotent groups

“…, a c ∈ G}. It has been proved in[2] that B c (G) is a characteristic subgroup of G, and that x ∈ B c (G) if and only if [xa 0 , g, a 1 , . .…”