Invariants in noncommutative dynamics Chirvasitu, Alexandru; Passer, Benjamin
Journal of functional analysis,
10/2019, Letnik:
277, Številka:
8
Journal Article
Recenzirano
Odprti dostop
When a compact quantum group H coacts freely on unital C⁎-algebras A and B, the existence of equivariant maps A→B may often be ruled out due to the incompatibility of some invariant. We examine the ...limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk–Ulam conjectures of Baum–Dąbrowski–Hajac. Among our results, we find that for certain finite-dimensional H, there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of H. This claim is in stark contrast to the case when H is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of H to be cleft as comodules over the Hopf algebra associated to H. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a θ-deformation procedure.
In this paper , we study the concept of the c-nilpotent multiplier of a pair of groups and prove that the c-nilpotent multipliers of perfect pairs of groups are isomorphic .Also, we prove an ...inequality for the order of the Schur multiplier of a pair of groups
Motivated by the application in geometric orthogonal codes (GOCs), Wang et al. introduced the concept of generalized perfect difference families (PDFs), and established the equivalence between GOCs ...and a certain type of generalized PDFs recently. Based on the relationship, we discuss the existence problem of generalized (n×m,K,1)-PDFs in this paper. By using some auxiliary designs such as semi-perfect group divisible designs and several recursive constructions, we prove that a generalized (n×m,{3,4},1)-PDF exists if and only if nm≡1(mod6). The existence of a generalized (n×m,{3,4,5},1)-PDF is also completely solved possibly except for a few values. As a consequence, some variable-weight perfect (n×m,K,1)-GOCs are obtained.
Let G be a group and α ϵ Aut(G). An α-commutator of elements x, y ϵ G is defined as x, yα = x-1y-1xyα. In 2015, Barzegar et al. introduced an α-commutator of elements of G and defined a new ...generalization of nilpotent groups by using the definition of α-commutators which is called an α-nilpotent group. They also introduced an α-commutator subgroup of G, denoted by Dα(G) which is a subgroup generated by all α-commutators. In 2016, an α-perfect group, a group that is equal to its α-commutator subgroup, was introduced by authors of this paper and the properties of such group was investigated. They proved some results on α-perfect abelian groups and showed that a cyclic group G of even order is not α-perfect for any α ϵ Aut(G). In this paper, we may continue our investigation on α-perfect groups and in addition to studying the relative perfectness of some classes of finite p-groups, we provide an example of a non-abelian α-perfect 2-group.
Let cd(G) be the set of the degrees of all complex irreducible characters of a finite group G. For a finite nonabelian simple group S and a positive integer k, let Sk be the direct product of k ...copies of S. In 2, we conjectured that all finite groups G with cd(G)=cd(Sk) are quasi perfect groups (that is; G′=G″) and hence nonsolvable groups. Then we proved that this conjecture holds for some sporadic simple groups as well as for some simple groups of Lie type (see 1 and 2). In this paper, we verify this conjecture for some alternating groups and for the simple groups Psp4(q)(q=2m≥2) and G22(q2)(q=32m+1≥27). Indeed, we show that if G is a finite group with cd(G)=cd(H), where H∈{A7k,S7k(k≥1),Psp4(q)k(q=2m≥2,k≥1),G22(q2)k(q=32m+1≥27,1≤k≤6560),A8k(1≤k≤5),S8,A9k,S9k,A10k,S10k(1≤k≤2)}, then G is a quasi perfect group.
This paper aims to introduce and explore the concept of Lie perfect multiplicative Lie algebras, with a particular focus on their connections to the central extension theory of multiplicative Lie ...algebras. The primary objective is to establish and provide proof for a range of results derived from Lie perfect multiplicative Lie algebras. Furthermore, the study extends the notion of Lie nilpotency by introducing and examining the concept of local nilpotency within multiplicative Lie algebras. The paper presents an innovative adaptation of the Hirsch-Plotkin theorem specifically tailored for multiplicative Lie algebras.
On Groups with Certain Proper FC-Subgroups Alkış, Og~uz; Arıkan, Ahmet; Arıkan, Aynur
Algebras and representation theory,
08/2022, Letnik:
25, Številka:
4
Journal Article
Recenzirano
Let
G
be a group. If for every proper normal subgroup
N
and element
x
of
G
with
N
〈
x
〉≠
G
,
N
〈
x
〉 is an
FC
-group, but
G
is not an
FC
-group, then we call
G
an
NFC
-
group
. In the present paper ...we consider the
NFC
-groups. We prove that every non-perfect
NFC
-group with non-trivial finite images is a minimal non-
FC
-group. Also we show that if G is a non-perfect
NFC
-group having no nontrivial proper subgroup of finite index, then
G
is a minimal non-
FC
-group under the condition “every Sylow
p
-subgroup is an
FC
-group for all primes
p
”. In the perfect case, we show that there exist locally nilpotent perfect
NFC
-
p
-groups which are not minimal non-
FC
-groups and also that McLain groups
M
(
ℚ
,
G
F
(
p
)
)
for any prime
p
contain such groups. We give a characterization for torsion-free case. We also consider the
p
-groups such that the normalizer of every element of order
p
is an
FC
-subgroup.
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