Totally imprimitive permutation groups with the cyclic-block property

Asar, A. O.

doi:10.1515/jgt.2010.043

Cited by 4 publications

(9 citation statements)

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“…In this section the finitary permutation p -group given in [4] and satisfying the cyclic-block property is described briefly for the convenience of the reader. This group is a subgroup of the example given in [23] by Wiegold.…”

Section: A Finitary Permutation Group With Cyclic-block Propertymentioning

confidence: 99%

“…If a perfect M N F -group exists, then it is a p-group for a prime p by [7, Theorem 2] and [14,Theorem], and it has a nontrivial representation in the group of finitary permutations on some infinite set by the characterizations given in [8,15]. (Some partial results in this direction are contained in [1][2][3][4][5]. )…”

Section: Introductionmentioning

confidence: 99%

“…Now suppose in addition that G satisfies the cyclic-block property. By [4,Lemma 2.2] two blocks for G are either disjoint or one is contained in the other one. This implies that G must be a p-group for a prime p .…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, Theorem 1.1 provides a short proof for [4, Theorem 1.2] (Corollary 1.3). (Another proof of [4,Theorem 1.2] is contained in [5,Theorem 1.6].) The group given in [4,Theorem 1.1] satisfies the cyclic-block property, it has an easily defined generating set, and all of its blocks of p -power size are easily described, but it is not known whether or not it contains an M N F C -subgroup.…”

Section: Introductionmentioning

confidence: 99%

“…(Another proof of [4,Theorem 1.2] is contained in [5,Theorem 1.6].) The group given in [4,Theorem 1.1] satisfies the cyclic-block property, it has an easily defined generating set, and all of its blocks of p -power size are easily described, but it is not known whether or not it contains an M N F C -subgroup. (This group satisfying the cyclic-block property is a transitive subgroup of the maximal p -subgroup, denoted by W here, of F Sym(N * ) constructed in [22]; see Proposition 2.1 for some properties of W .)…”

Section: Introductionmentioning

confidence: 99%

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Permutation groups with cyclic-block property and $MNFC$-groups

Asar¹

2017

Turk J Math

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This work continues the investigation of perfect locally finite minimal non-F C -groups in totally imprimitive permutation p -groups. At present, the class of totally imprimitive permutation p -groups satisfying the cyclic-block property is known to be the only class of p -groups having common properties with a hypothetical minimal non-F Cgroup, because a totally imprimitive permutation p -group satisfying the cyclic-block property cannot be generated by a subset of finite exponent and every non-F C -subgroup of it is transitive, which are the properties satisfied by a minimal non-F C -group. Here a sufficient condition is given for the nonexistence of minimal non-F C -groups in this class of permutation groups. In particular, it is shown that the totally imprimitive permutation p -group satisfying the cyclic-block property that was constructed earlier and its commutator subgroup cannot be minimal non-F C -groups. Furthermore, some properties of a maximal p -subgroup of the finitary symmetric group on N * are obtained.

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Section: A Finitary Permutation Group With Cyclic-block Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Permutation groups with cyclic-block property and $MNFC$-groups

Asar¹

2017

Turk J Math

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On Groups with Certain Proper FC-Subgroups

Alkış

Arikan

2021

Algebr Represent Theor

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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 F C-group, but G is not an F C-group, then we call G an NF C-group. In the present paper we consider the NF C-groups. We prove that every non-perfect NF Cgroup with non-trivial finite images is a minimal non-F C-group. Also we show that if G is a non-perfect NF C-group having no nontrivial proper subgroup of finite index, then G is a minimal non-F C-group under the condition "every Sylow p-subgroup is an F C-group for all primes p". In the perfect case, we show that there exist locally nilpotent perfect NF C-pgroups which are not minimal non-F C-groups and also that McLain groups M(Q, GF (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 F Csubgroup.

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N-barely transitive permutation groups

Alkış

Arikan

2021

Ricerche mat

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Totally imprimitive permutation groups with the cyclic-block property

Cited by 4 publications

References 4 publications

Permutation groups with cyclic-block property and $MNFC$-groups

Permutation groups with cyclic-block property and $MNFC$-groups

On Groups with Certain Proper FC-Subgroups

N-barely transitive permutation groups

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