Shintani descent, simple groups and spread

Harper, Scott

doi:10.1016/j.jalgebra.2021.02.021

Cited by 6 publications

(17 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us explain how the Shintani map relates to closed subgroups of X. This is studied comprehensively in [26], but apart from in a few instances, all we need is a simpler result. We have chosen to give the proof since it is short and enlightening.…”

Section: Shintani Descentmentioning

confidence: 99%

“…We take this section to introduce our notation for the almost simple groups of Lie type. Our notation is consistent with [26], to which we will refer later.…”

Section: Groups Of Lie Typementioning

confidence: 99%

“…For instance, if X = PGL n (F p ) and ϕ is the standard Frobenius map, then for divisors i of f we have a bijection between the conjugacy classes in PGL n (p f )ϕ i and in PGL n (p i ). This bijection preserves important structure, including information about maximal overgroups (see [26,Theorem 4]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Spread of Almost Simple Classical Groups

Harper¹

2021

Lecture Notes in Mathematics

View full text Add to dashboard Cite

show abstract

Section: Shintani Descentmentioning

confidence: 99%

“…We take this section to introduce our notation for the almost simple groups of Lie type. Our notation is consistent with [26], to which we will refer later.…”

Section: Groups Of Lie Typementioning

confidence: 99%

See 1 more Smart Citation

The Spread of Almost Simple Classical Groups

Harper¹

2021

Lecture Notes in Mathematics

View full text Add to dashboard Cite

show abstract

“…A key tool to do this is the theory of Shintani descent, which we outline here following the work of Kawanaka [58,Section 2]. We also refer to [19,Section 2] and the more recent improvements of Harper on the theory of Shintani descent [52].…”

Section: Dropping the Maximalitymentioning

confidence: 99%

Normal $2$-coverings of the finite simple groups and their generalizations

Bubboloni¹,

Spiga²,

Weigel³

2022

Preprint

View full text Add to dashboard Cite

Given a finite group G, we say that G has weak normal covering number γw(G) if γw(G) is the smallest integer with G admitting proper subgroups H 1 , . . . , H γw (G) such that each element of G has a conjugate in H i , for some i ∈ {1, . . . , γw(G)}, via an element in the automorphism group of G.We prove that the weak normal covering number of every non-abelian simple group is at least 2 and we classify the non-abelian simple groups attaining 2. As an application, we classify the non-abelian simple groups having normal covering number 2. We also show that the weak normal covering number of an almost simple group is at least two up to one exception.We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.

show abstract

“…Thus, all that remains is to prove the theorem when α is non-trivial, and when we are not in the case where S ∈ {A ℓ (q), D ℓ (q), E 6 (q)} and α is a graph automorphism of S. Then α is the restriction to S = O p ′ (X σ ) of a Steinberg endomorphism α of X. In these cases, the theory of Shintani descent (see for example, [13]) gives us what we need. Indeed, by [13, Theorem 2.1 and Lemma 2.16], the coset Sα contains a unipotent element x with the property that |C Xσ (x)| = |C Xσ 1 (x 1 )|, where…”

Section: The Proof Of Theoremmentioning

confidence: 99%

On the proportion of $p$-elements in a finite group, and a modular Jordan type theorem

Tracey¹

2021

Preprint

View full text Add to dashboard Cite

In 1878, Jordan proved that if a finite group G has a faithful representation of dimension n over C, then G has a normal abelian subgroup with index bounded above by a function of n. The same result fails if one replaces C by a field of positive characteristic, due to the presence of large unipotent and/or Lie type subgroups. For this reason, a long-standing problem in group and representation theory has been to find the "correct analogue" of Jordan's theorem in characteristic p > 0. Progress has been made in a number of different directions, most notably by Brauer and Feit in 1966;by Collins in 2008; and by Larsen and Pink in 2011. With a 1968 theorem of Steinberg in mind (which shows that a significant proportion of elements in a simple group of Lie type are unipotent), we prove in this paper that if a finite group G has a faithful representation over a field of characteristic p, then a significant proportion of the elements of G must have p-power order. We prove similar results for permutation groups, and present a general method for counting p-elements in finite groups. All of our results are best possible.

show abstract

Shintani descent, simple groups and spread

Cited by 6 publications

References 36 publications

The Spread of Almost Simple Classical Groups

The Spread of Almost Simple Classical Groups

Normal $2$-coverings of the finite simple groups and their generalizations

On the proportion of $p$-elements in a finite group, and a modular Jordan type theorem

Contact Info

Product

Resources

About