The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. ...Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if F is a subfield of a local field of characteristic ≠2, then the special upper triangular group ST+(n,F) is minimal precisely when the special linear group SL(n,F) is. We provide criteria for the minimality (and total minimality) of SL(n,F) and ST+(n,F), where F is a subfield of C. Let Fπ and Fc be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for A∈{Fπ,Fc}: A is finite; ∏Fn∈ASL(Fn−1,Q(i)) is minimal, where Q(i) is the Gaussian rational field; and ∏Fn∈AST+(Fn−1,Q(i)) is minimal. Similarly, denote by Mπ and Mc the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let B∈{Mπ,Mc}. Then the following conditions are equivalent: B is finite; ∏Mp∈BSL(Mp+1,Q(i)) is minimal; and ∏Mp∈BST+(Mp+1,Q(i)) is minimal.
In relation to Itzkowitz’s problem 5, we show that a c-bounded P-group is balanced if and only if it is functionally balanced.We prove that for an arbitrary P-group, being functionally balanced is ...equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform P-space. In particular, we show that this group is (functionally) balanced precisely when its subsets B
, consisting of words of length at most n, are all (resp., functionally) balanced.
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian ...groups,
(2009), 7, 3401–3434.
Later, this result was extended to all abelian groups D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups,
(2016), 612–653 and, recently, to all torsion finitely quasihamiltonian groups A. Giordano Bruno and F. Salizzoni, Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups,
(2020), 5, 831–846.
In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms,
(2017), 4, 763–774.
Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.
C-minimal topological groups Xi, Wenfei; Shlossberg, Menachem
Forum mathematicum,
11/2021, Letnik:
33, Številka:
6
Journal Article
Recenzirano
In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups
. We show that a locally compact c-minimal connected group is compact. Using a ...well-known theorem of P. Hall and C. R. Kulatilaka,
A property of locally finite groups,
J. Lond. Math. Soc. 39 1964, 235–239 and a characterization of a certain class of Lie groups, due to S. K. Grosser and W. N. Herfort,
Abelian subgroups of topological groups,
Trans. Amer. Math. Soc. 283 1984, 1, 211–223, we prove that a c-minimal locally solvable Lie group is compact.
It is shown that a topological group
is c-(totally) minimal if and only if
has a compact normal subgroup
such that
is c-(totally) minimal.
Applying this result, we prove that a locally compact group
is c-totally minimal if and only if its connected component
is compact and
is c-totally minimal.
Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering D. Dikranjan and M. Megrelishvili,
Minimality conditions in topological groups,
Recent Progress in General Topology. III,
Atlantis Press, Paris 2014, 229–327, Question 3.10 (b), we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.
We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact ...open normal subgroup of G. For continuous endomorphisms ϕ:G→G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions.
Hereditarily minimal topological groups Xi, Wenfei; Dikranjan, Dikran; Shlossberg, Menachem ...
Forum mathematicum,
05/2019, Letnik:
31, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We study locally compact groups having all subgroups minimal. We call such groups
.
In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups
of
...-adic integers.
We extend Prodanov’s theorem to the non-abelian case at several levels.
For infinite hypercentral (in particular, nilpotent) locally compact groups, we show that the hereditarily minimal ones remain the same as in the abelian case.
On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that, in particular, they are always compact and metabelian.
The proofs involve the (hereditarily) locally minimal groups, introduced similarly.
In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group
is hereditarily locally minimal, where
is the multiplicative group of non-zero
-adic numbers acting on the first component by multiplication.
Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.
We show that the Heisenberg type group HX=(Z2⊕V)⋋V⁎, with the discrete Boolean group V:=C(X,Z2), canonically defined by any Stone space X, is always minimal. That is, HX does not admit any strictly ...coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean G there exists a (resp., locally compact) non-archimedean minimal group M such that G is a group retract of M. For discrete groups G the latter was proved by S. Dierolf and U. Schwanengel (1979) 6. We unify some old and new characterization results for non-archimedean groups.
It is still open whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning the Mersenne numbers is also unsolved. Extending some ...results from 9, we characterizethe the Fermat primes and the Mersenne primes in terms of topological minimality of some matrix groups. This is done by showing, among other things, that if \(\Bbb{F}\) is a subfield of a local field of characteristic \(\neq 2,\) then the special upper triangular group \(\operatorname{ST^+}(n,\Bbb{F})\) is minimal precisely when the special linear group \(\operatorname{SL}(n,\Bbb{F})\) is. We provide criteria for the minimality (and total minimality) of \(\operatorname{SL}(n,\Bbb{F})\) and \(\operatorname{ST^+}(n,\Bbb{F}),\) where \(\Bbb{F}\) is a subfield of \(\Bbb{C}.\) Let \(\mathcal F_\pi\) and \(\mathcal F_c \) be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for \(\mathcal{A}\in \{\mathcal F_\pi, \mathcal F_c\}:\) \(\bullet \ \mathcal{A}\) is finite; \(\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{SL}(F_n-1, \Bbb{Q}(i))\) is minimal, where \(\Bbb{Q}(i)\) is the Gaussian rational field; \(\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{ST^+}(F_n-1, \Bbb{Q}(i))\) is minimal. Similarly, denote by \(\mathcal M_\pi\) and \(\mathcal M_c \) the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let \(\mathcal{B}\in\{ \mathcal M_\pi, \mathcal M_c\}.\) Then the following conditions are equivalent: \(\bullet \ \mathcal B\) is finite; \(\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{SL}(M_p+1, \Bbb{Q}(i))\) is minimal; \(\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{ST^+}(M_p+1, \Bbb{Q}(i))\) is minimal.
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in 4. Later, this result was extended to all abelian groups 3 and, recently, to all torsion ...finitely quasihamiltonian groups 7. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails 8. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.
We introduce two minimality properties of subgroups in topological groups. A subgroup \(H\) is a key subgroup (co-key subgroup) of a topological group \(G\) if there is no strictly coarser Hausdorff ...group topology on \(G\) which induces on \(H\) (resp., on the coset space \(G/H\)) the original topology. Every co-minimal subgroup is a key subgroup while the converse is not true. Every locally compact co-compact subgroup is a key subgroup (but not always co-minimal). Any relatively minimal subgroup is a co-key subgroup (but not vice versa). Extending some results concerning the generalized Heisenberg groups, we prove that the center ("corner" subgroup) of the upper unitriangular group \(\mathrm{UT(n,K)}\), defined over a commutative topological unital ring \(K\), is a key subgroup. Every "non-corner" 1-parameter subgroup \(H\) of \(\mathrm{UT(n,K)}\) is a co-key subgroup. We study injectivity property of the restriction map $$r_H \colon \mathcal{T}_{\downarrow}(G) \to \mathcal{T}_{\downarrow}(H), \ \sigma \mapsto \sigma|_H$$ and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup \(H\), where \(\mathcal{T}_{\downarrow}(G)\) is the semilattice of coarser Hausdorff group topologies on \(G\).
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