Abstract A Grigorchuk–Gupta–Sidki (GGS)-group is a subgroup of the automorphism group p -regular rooted tree for an odd prime , generated by one and directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups infinite index. Here, we extend result to nontorsion GGS-groups, which include weakly regular branch, but GGS-group.

1. Resonance of PΣn Let Fn be the free group generated by x1, . . . , xn. The basis-conjugating automorphism group, or pure symmetric automorphism group, is the group PΣn of all automorphisms of Fn which send each generator xi to a conjugate of itself. Results of Dahm [4] and Goldsmith [6] imply that this group may also be realized as the “group of loops,” the group of motions of a collection o...

We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization V of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Joh...

Let F be a perfect field and M(F ) the nonassociative simple Moufang loop consisting of the units in the (unique) split octonion algebra O(F ) modulo the center. Then Aut(M(F )) is equal to G2(F )o Aut(F ). In particular, every automorphism of M(F ) is induced by a semilinear automorphism of O(F ). The proof combines results and methods from geometrical loop theory, groups of Lie type and compo...

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is...