For positive integers
n
and
k
, with
n
≥
4
, let
F
n
be the free group of rank
n
and let
G
n
,
k
=
F
n
/
γ
3
(
F
n
′
)
F
n
″
,
k
F
n
. We show that for sufficiently large
n
, the automorphism group
...Aut
(
G
n
,
k
)
of
G
n
,
k
is generated by the tame automorphisms and one more non-tame automorphism.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We prove that in every variety of $G$-groups, every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize Theorem G of J. ...Algebra, 219 (1999) 16--79. As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{\omega}$-compact, they are geometrically equivalent.
Let m, n be positive integers, v a multilinear commutator word and w=vm. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of ...all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result remains valid with m allowed to be an arbitrary positive integer. In this direction, we show that if m, n are positive integers, u a multilinear commutator word and v the product of 896 u-words, then the class of all groups G in which the vm-values are n-Engel and the verbal subgroup u(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem C).
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We present a general criterion under which the equality var(
A
wr
B
) = var(
A
)var(
B
) holds for finite groups
A
and
B
. This generalizes some known results in this direction and continues our ...previous research
J. Alg.
,
313
, No. 2, 455-458 (2007) on varieties generated by wreath products of Abelian groups. The classification is based on the techniques developed by A. L. Shmel’kin, R. Burns, etc., who used critical groups, verbal wreath products, and Cross properties for studying critical groups in nilpotent-by-Abelian varieties.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A variety of groups does not contain all metabelian groups if and only if there is an absolute bound for the nilpotency classes of powerful
p
-groups in the given variety. Similarly, a variety ...contains only finitely many finite
p
-groups of any given coclass if and only if not every group that is an extension of an abelian group by an elementary abelian
p
-group belongs to that variety.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We propose a classification of all cases when, for a nilpotent group of finite exponent and an abelian group , the variety generated by their wreath product is equal to the product of the varieties ...and generated by and . This generalizes some results in the literature concerning the same problem for more restricted types of groups. This classification extends our earlier work on varieties generated by wreath products of abelian groups and wreath products of finite groups.
We study the variety generated by cartesian and direct wreath products of arbitrary sets
X
and
Y
of abelian groups. In particular, we give a classification of the cases when that variety is equal to ...the product variety
var
(
X
)
var
(
Y
)
. This criterion is a wide generalization of the theorems of Higman and Houghton about the varieties generated by wreath products of cycles, of a few other known examples about the varieties generated by wreath products of abelian groups (and of sets of abelian groups), and also of our recent research about the varieties generated by wreath products of abelian groups.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Let $G$ be a finite nilpotent group of class three whose proper subgroups and proper quotients are nilpotent of class at most two. We show that $G$ is either a 2-generated $p$-group or a 3-generated ...3-group. In the first case the groups of maximal order with respect to a given exponent are all isomorphic except in the cases where $p=2$ and $\mathrm {exp}(G)= 2^r$, $r\geq 4$. If $G$ is 3-generated, then we show that there is a unique group of maximal order and exponent 3; but a similar result is not valid for exponent 9.
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