On Subgroups of Free Burnside Groups of Large Odd Exponent
نویسنده
چکیده
We prove that every noncyclic subgroup of a free m-generator Burnside group B(m, n) of odd exponent n ≫ 1 contains a subgroup H isomorphic to a free Burnside group B(∞, n) of exponent n and countably infinite rank such that for every normal subgroup K of H the normal closure 〈K〉B(m,n) of K in B(m, n) meets H in K. This implies that every noncyclic subgroup of B(m, n) is SQ-universal in the class of groups of exponent n. A group G is called SQ-universal if every countable group is isomorphic to a subgroup of a quotient of G. One of classical embedding theorems proved by Higman, B. Neumann, H. Neumann in [HNN49] states that every countable group G embeds in a 2-generator group or, equivalently, a free group F2 of rank 2 is SQ-universal. Recall that the proof of this theorem makes use of the following natural definition. A subgroup H of a group G is called a Q-subgroup if for every normal subgroup K of H the normal closure 〈K〉 of K in G meets H in K, i.e., 〈K〉 ∩ H = K. For example, factors G1, G2 of the free product G1 ∗ G2 or the direct product G1 ×G2 are Q-subgroups of G1 ∗G2 or G1 ×G2, respectively. In particular, a free group Fm of rank m > 1, where m = ∞ means countably infinite rank, contains a Q-subgroup isomorphic to Fk for every k ≤ m. On the other hand, it is proved in [HNN49] that the subgroup 〈abababababababab | i = 1, 2, . . . 〉 of F2 = F2(a, b) is a Q-subgroup of F2 isomorphic to F∞ and freely generated by indicated elements. In [NN59] B. Neumann and H. Neumann found simpler generators and proved that 〈[bab, a] | i = 1, 2, . . . 〉, where [x, y] = xyxy is the commutator of x and y, is a Q-subgroup of F2 isomorphic to F∞ and freely generated by indicated elements. It is obvious that the property of being a Q-subgroup is transitive. Therefore, a group G contains a Q-subgroup isomorphic to F∞ if and only if G contains a Q-subgroup isomorphic to Fm, where m ≥ 2. Ol’shanskii [O95] proved that any nonelementary subgroup of a hyperbolic group G (in particular, G = Fm) contains a Q-subgroup isomorphic to F2. In particular, if G is a nonelementary hyperbolic group then G is SQ-universal. It follows from an embedding theorem of Obraztsov (see Theorem 35.1 in [O89]) that any countable group of odd exponent n ≫ 1 embeds in a 2-generator group of exponent n and so a free 2-generator Burnside group B(2, n) = F2/F n 2 of exponent n is SQ-universal in the class of groups of exponent n. Interestingly, the proof of this theorem has nothing to do with free Q-subgroups of the Burnside group B(2, n) and does not imply the existence of such subgroups in B(2, n). Ol’shanskii and Sapir proved in [OS02] (among many other things) that for odd n ≫ 1 the group B(m, n) with some m = m(n) does contain Q-subgroups 2000 Mathematics Subject Classification. Primary 20E07, 20F05, 20F50. Supported in part by NSF grant DMS 00-99612.
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