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Some questions on quasinilpotent groups and related classes.

M.J. Iranzo, J. Medina, F. Pérez-Monasor (2002)

Revista Matemática Iberoamericana

In this paper we will prove that if G is a finite group, X a subnormal subgroup of X F*(G) such that X F*(G) is quasinilpotent and Y is a quasinilpotent subgroup of NG(X), then Y F*(NG(X)) is quasinilpotent if and only if Y F*(G) is quasinilpotent. Also we will obtain that F*(G) controls its own fusion in G if and only if G = F*(G).

Some regular quasivarieties of commutative binary modes

K. Matczak, Anna B. Romanowska (2014)

Commentationes Mathematicae Universitatis Carolinae

Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety 𝒱 of groupoids is the smallest variety containing 𝒱 and the variety 𝒮 of semilattices. Its quasiregularization is the smallest quasivariety containing 𝒱 and 𝒮 . In an earlier paper the authors described the lattice of quasivarieties of cancellative commutative binary modes, i.e. idempotent commutative and entropic (or medial) groupoids. They are all irregular...

Some relations on the lattice of varieties of completely regular semigroups

Mario Petrich (2002)

Bollettino dell'Unione Matematica Italiana

On the lattice L C R of varieties of completely regular semigroups considered as algebras with the binary multiplication and unary inversion within maximal subgroups, we study the relations K l , K , K r , T l , T , T r , C and L . Here K is the kernel relation, T is the trace relation, T l and T r are the left and the right trace relations, respectively, K p = K T p for p l , r , C is the core relation and L is the local relation. We give an alternative definition for each of these relations P of the form U P V U P ~ = V P ~ ( U , V L ( C R ) ) , for some subclasses P ~ of C R ....

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