411 research outputs found
Beauville structures in finite p-groups
We study the existence of (unmixed) Beauville structures in finite
-groups, where is a prime. First of all, we extend Catanese's
characterisation of abelian Beauville groups to finite -groups satisfying
certain conditions which are much weaker than commutativity. This result
applies to all known families of -groups with a good behaviour with respect
to powers: regular -groups, powerful -groups and more generally potent
-groups, and (generalised) -central -groups. In particular, our
characterisation holds for all -groups of order at most , which allows
us to determine the exact number of Beauville groups of order , for , and of order , for . On the other hand, we determine which
quotients of the Nottingham group over are Beauville groups, for
an odd prime . As a consequence, we give the first explicit infinite family
of Beauville -groups, and we show that there are Beauville -groups of
order for every
Outer commutator words are uniformly concise
We prove that outer commutator words are uniformly concise, i.e. if an outer
commutator word w takes m different values in a group G, then the order of the
verbal subgroup w(G) is bounded by a function depending only on m and not on w
or G. This is obtained as a consequence of a structure theorem for the subgroup
w(G), which is valid if G is soluble, and without assuming that w takes
finitely many values in G. More precisely, there is an abelian series of w(G),
such that every section of the series can be generated by values of w all of
whose powers are also values of w in that section. For the proof of this latter
result, we introduce a new representation of outer commutator words by means of
binary trees, and we use the structure of the trees to set up an appropriate
induction
On the congruence subgroup property for GGS-groups
We show that all GGS-groups with non-constant defining vector satisfy the
congruence subgroup property. This provides, for every odd prime , many
examples of finitely generated, residually finite, non-torsion groups whose
profinite completion is a pro- group, and among them we find torsion-free
groups. This answers a question of Barnea. On the other hand, we prove that the
GGS-group with constant defining vector has an infinite congruence kernel and
is not a branch group.Comment: v2 incorporates referee suggestions (final version
Hausdorff dimension in -analytic profinite groups
We study the Hausdorff dimension of R-analytic subgroups in an R-analytic
profinite group, where R is a pro-p ring whose asso- ciated graded ring is an
integral domain. In particular, we prove that the set of such Hausdorff
dimensions is a finite subset of the rational numbers.Comment: 7 page
A restriction on centralizers in finite groups
For a given m>=1, we consider the finite non-abelian groups G for which
|C_G(g):|<=m for every g in G\Z(G). We show that the order of G can be
bounded in terms of m and the largest prime divisor of the order of G. Our
approach relies on dealing first with the case where G is a non-abelian finite
p-group. In that situation, if we take m=p^k to be a power of p, we show that
|G|<=p^{2k+2} with the only exception of Q_8. This bound is best possible, and
implies that the order of G can be bounded by a function of m alone in the case
of nilpotent groups
Telecom submarine cables, a business compatible with the environment protection
All playing sgents in marine operation; telecoms, cable manufactures and cableship operators, are more and more involved in making their increasingly watched and regulated activities to be compatible with sustainable and live oceans.Peer Reviewe
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