Groups with automorphisms inverting most elements

Liebeck, Hans; MacHale, Des

doi:10.1007/bf01142582

Cited by 37 publications

(56 citation statements)

References 6 publications

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“…It remains to show that G is of type III in Theorem 3.1. This is highly non-trivial, but the argument parallels entirely that in Section 4 of [LM1], with very minor modifications. We thus omit further details.…”

Section: In Particular If (|G| 3) = 1 Then It Suffices For G To Havmentioning

confidence: 53%

“…In what is probably the most significant paper in this area, Liebeck and MacHale [LM1] provided a concise classification of those groups admitting an automorphism which inverts more than half their elements. MacHale and the author [HM] extended this classification to include groups admitting an automorphism which inverts exactly half the group elements, but already here the classification is considerably more detailed.…”

Section: Introductionmentioning

confidence: 99%

“…The methods introduced in [LM1] provide the basis for much of the subsequent investigations in the papers cited above. Let n ∈ {−1, 2}.…”

Section: Introductionmentioning

confidence: 99%

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Finite Groups With an Automorphism Cubing a Large Fraction of Elements

Hegarty¹

2009

Mathematical Proceedings of the Royal Irish Academy

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ABSTRACT. We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A 5 shows that this result is best-possible.Both our main findings closely parallel results of prevous authors on finite groups possessing an automorphism which inverts many group elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.

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Section: In Particular If (|G| 3) = 1 Then It Suffices For G To Havmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

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Finite Groups With an Automorphism Cubing a Large Fraction of Elements

Hegarty¹

2009

Mathematical Proceedings of the Royal Irish Academy

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“…Related problems have been investigated by various authors. For example, Liebeck and MacHale [7] classify the finite groups in which more than half of the elements are inverted by some automorphism of the group, extending earlier work of Manning and Miller (see [8,9], for example). All such groups are soluble, and the aforementioned theorem of Wall follows as a corollary.…”

Section: Introductionmentioning

confidence: 77%

“…Moreover, if y ∈ N has odd order then n j ∈ C N (y) for all 1 ≤ j ≤ m, hence y ∈ Z (N ) since m > 2|N |/3. Therefore, the set of elements of [7] On the number of prime order subgroups of finite groups 335 The next lemma provides rather accurate bounds on i 2 (G), i 3 (G) and |G| in the case where G is a simple group of Lie type. In view of the isomorphisms G 2 (2) ∼ = U 3 (3) and 2 Table 2 we regard G 2 (2) and 2 G 2 (3) as classical groups.…”

Section: Preliminariesmentioning

confidence: 99%

On the Number of Prime Order Subgroups of Finite Groups

Burness¹,

Scott

2009

J. Aust. Math. Soc.

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On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups

Yan

Spiga

2020

Journal of Graph Theory

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In this paper, we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let A be an abelian group and let ι be the automorphism of A defined by a a = ι −1 , for every a A ∈. A Cayley graph A S Cay(,) is said to have an automorphism group as small as possible if A S A ι Aut(Cay(,)) = , 〈 〉. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley digraphs. These results are used for the asymptotic enumeration of bipartite Cayley digraphs and graphs over abelian groups.

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Groups with automorphisms inverting most elements

Cited by 37 publications

References 6 publications

Finite Groups With an Automorphism Cubing a Large Fraction of Elements

Finite Groups With an Automorphism Cubing a Large Fraction of Elements

On the Number of Prime Order Subgroups of Finite Groups

On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups

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