Trivial Set-Stabilizers in Finite Permutation Groups

Gluck, David

doi:10.4153/cjm-1983-005-2

Cited by 63 publications

(51 citation statements)

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“…Then K/A is the direct product of a group isomorphic to the alternating group A 4 and an extension of a completely reducible nilpotent subgroup of SL (2, 5) We can again consider just the case a = 1. By Lemma 2.1(6) and (7), G is the central product of groups G 1 (C p−1 E 1 ).SL (2,3) where E 1 is extraspecial of order 3 3 and G 2 (C p−1 E 2 ).D, with E 2 extraspecial, |E 2 | = 2 5 and D isomorphic to either S 3 S 2 or a Frobenius group of order 20 (see [14, §8.1]…”

Section: Lemma 33 Let R Be a Solvable Group U An Elementary Abeliamentioning

confidence: 99%

Large orbits in coprime actions of solvable groups

Dolfi¹

2008

Trans. Amer. Math. Soc.

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Section: Lemma 33 Let R Be a Solvable Group U An Elementary Abeliamentioning

confidence: 99%

Large orbits in coprime actions of solvable groups

Dolfi¹

2008

Trans. Amer. Math. Soc.

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“…Our approach roughly parallels that of [6]. This paper and [5] contain new applications of the results and methods of [6, 7,1,2]. …”

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Primes dividing character degrees and character orbit sizes

Gluck¹

1987

Proc. Amer. Math. Soc.

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Abstract. We consider an abelian group A which acts faithfully and coprimely on a solvable group G. We show that some A -orbit on Irr(G) must have cardinality divisible by almost half the primes in ir(A). As a corollary, we improve a recent result of I. M. Isaacs concerning the maximum number of primes dividing any one character degree of a solvable group.A recent result of I. M. Isaacs [5, Corollary 4.3] relates the maximum number of primes dividing any one irreducible character degree of a solvable group G to the number of primes dividing all the character degrees of G taken together. Here we considerably strengthen the bound in [5, Corollary 4.3] by proving a result on character orbit sizes in coprime actions.We consider an abelian group A which acts faithfully and coprimely on a solvable group G. We show that some ,4-orbit on Irr(G) must have cardinality divisible by almost half the primes in tr(A). Our approach roughly parallels that of [6]. This paper and [5] contain new applications of the results and methods of [6, 7,1,2].

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“…On the other hand, the special case of distinguishing number two had been considered previously for permutation groups. For example, a theorem by Gluck [13] states that D(A, X) = 2 whenever |A| is odd, and this preceded [2] by ten years. Albertson and Collins were motivated by a long-standing recreational mathematics question on coloring beads on a necklace (or keys on a key chain) so as to destroy the dihedral symmetry: two colors suffice if the number of beads is at least 6 and one of the colors is used on only three beads.…”

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confidence: 99%

Motion and distinguishing number two

Conder

Tucker

2011

Ars Math. Contemp.

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A group A acting faithfully on a finite set X is said to have distinguishing number two if there is a proper subset Y whose (setwise) stabilizer is trivial. The motion of A acting on X is defined as the largest integer k such that all non-trivial elements of A move at least k elements of X. The Motion Lemma of Russell and Sundaram states that if the motion is at least 2 log 2 |A|, then the action has distinguishing number two. When X is a vector space, group, or map, the Motion Lemma and elementary estimates of the motion together show that in all but finitely many cases, the action of Aut(X) on X has distinguishing number two. A new lower bound for the motion of any transitive action gives similar results for transitive actions with restricted point-stabilizers. As an instance of what can happen with intransitive actions, it is shown that if X is a set of points on a closed surface of genus g, and |X| is sufficiently large with respect to g, then any action on X by a finite group of surface homeomorphisms has distinguishing number two.

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Trivial Set-Stabilizers in Finite Permutation Groups

Cited by 63 publications

References 2 publications

Large orbits in coprime actions of solvable groups

Large orbits in coprime actions of solvable groups

Primes dividing character degrees and character orbit sizes

Motion and distinguishing number two

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