A permutational property of groups

Curzio, Mario; Longobardi, Patrizia; Maj, Mercede; Robinson, Derek J. S.

doi:10.1007/bf01229319

Cited by 57 publications

(22 citation statements)

References 3 publications

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“…The class of groups of this type was shown in [3] to consist exactly of the finite-by-abelian-by-finite groups. For the recent results and references on this extensively studied class of groups, we refer to [1].…”

Section: Pi Semigroup Algebras Of Linear Semigroupsmentioning

confidence: 99%

“…Proof. (1) GL has ¿?m and hence, [3], has a normal subgroup H of finite index such that the commutator subgroup (//, H) is finite. By [8,Proposition 7.3], Gc has no closed subgroups of finite index.…”

Section: Pi Semigroup Algebras Of Linear Semigroupsmentioning

confidence: 99%

“…[15]. On the other hand, K[S] was shown to be a Pi-algebra whenever 5 is a finitely generated semigroup (satisfying 9a ) of one of the following types: periodic [20], cancellative [11], 0-simple [3,5], inverse, or a Rees factor semigroup of free semigroup, cf. [12].…”

Section: Pi Semigroup Algebras Of Linear Semigroupsmentioning

confidence: 99%

“…It is straightforward that S has 3P whenever K[S] satisfies a polynomial identity. However the converse fails even for groups in view of [3] and the characterization of PI group algebras, cf. [15].…”

Section: Introductionmentioning

confidence: 99%

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PI semigroup algebras of linear semigroups

Okniński¹,

Putcha²

1990

Proc. Amer. Math. Soc.

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Abstract.It is well-known that if a semigroup algebra K[S] over a field K satisfies a polynomial identity then the semigroup 5 has the permutation property. The converse is not true in general even when S is a group. In this paper we consider linear semigroups 5 Ç Jfn(F) having the permutation property. We show then that K[S] has a polynomial identity of degree bounded by a fixed function of n and the number of irreducible components of the Zariski closure of 5 .A semigroup S is said to have the property âBm, m > 2, if for every ax, ... , ame S, there exists a non-trivial permutation a such that ax... am = aa(X) ■•■ao(m) • S has the permutation property 3° if S satisfies &>m for some m > 2. The class of groups of this type was shown in [3] to consist exactly of the finite-by-abelian-by-finite groups. For the recent results and references on this extensively studied class of groups, we refer to [1]. The above description of groups satisfying 3° was extended to cancellative semigroups in [11], while a study of regular semigroups with this property was begun in [6].In connection with the corresponding semigroup algebras K [S] over a field K, the problem of the relation between the property ¿P for 5 and the PIproperty for K[S] attracted the attention of several authors. It is straightforward that S has 3P whenever K[S] satisfies a polynomial identity. However the converse fails even for groups in view of [3] and the characterization of PI group algebras, cf. [15]. On the other hand, K[S] was shown to be a Pi-algebra whenever 5 is a finitely generated semigroup (satisfying 9a ) of one of the following types: periodic [20], cancellative [11], 0-simple [3, 5], inverse, or a Rees factor semigroup of free semigroup, cf. [12]. However, a finitely generated regular semigroup S with two non-zero ^-classes having 9° but with K[S] not being PI was constructed in [12].The main result of this paper is that if S is a linear semigroup satisfying 3P, then K[S] is PI for any field K. In the course of the proof, we obtain a structural description of a strongly 7r-regular semigroup of this type. The basic technique is to consider the Zariski closure S of S. Then S is a linear

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Section: Pi Semigroup Algebras Of Linear Semigroupsmentioning

confidence: 99%

Section: Pi Semigroup Algebras Of Linear Semigroupsmentioning

confidence: 99%

Section: Pi Semigroup Algebras Of Linear Semigroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

PI semigroup algebras of linear semigroups

Okniński¹,

Putcha²

1990

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Extensions of problems of this type are to be found in [7] and [6]. The notion of commutativity was extended to rewritable products in [4] with a complete description obtained in [5]. Detailed study of rewritable groups may be found in papers by R. Blyth [1] and [2].…”

Section: Introductionmentioning

confidence: 99%