On the depth of subgroups and group algebra extensions

We determine the combinatorial depth of certain subgroups of simple Suzuki groups Sz(q), among others, the depth of their maximal subgroups. We apply these results to determine the ordinary depth of these subgroups.L. H ÉTHELYI, E. HORV ÁTH, AND F. PET ÉNYIIn [1] the depth of a partially ordered set X is introduced, as the length of a largest chain in X, and δ is defined as the depth of U ∞ (H, G). The symbol δ * denotes the smallest positive integer k such that Core G (H) can be written as the intersection of k conjugates of H, and δ * denotes the smallest positive integer k such that the intersection of any k distinct conjugates of H is equal to Core G (H).Theorem 1.2. [1, Thm 3.11] Let G be a finite group, H a subgroup of G andNow let us remind the reader to the notion of ordinary depth of a subgroup H in the finite group G. We say that the depth of the group algebra inclusionFurthermore CH is said to have depth 2n + 1 in CG if the same assertion holds for CH − CH-bimodules. Finally CH has depth 1 in CG if CG is isomorphis to a direct summand of ⊕ a i=1 CH as CH − CH bimodules. The minimal depth of group algebra inclusion CH ⊆ CG is called (minimal) ordinary depth of H in G, which we denote by d(H, G). This is well defined.This depth can be obtained from the so called inclusion matrix M . If χ 1 , . . . , χ s are all irreducible characters of G and ψ 1 , . . . , ψ r are all irreducible characters of H, then m i,j := (ψ G i , χ j ). The "powers" of M are defined by M 2l := M 2l−1 M T and M 2l+1 := M 2l M The odinary depth d(H, G) can be obtained as the smallest integer n such that M n+1 ≤ aM n−1 . This is well defined. The results on characters in [2] help to determine d(H, G). Two irreducible characters α, β ∈ Irr(H) are related, α ∼ β, if they are constituents of some χ H , for χ ∈ Irr(G). The distance d(α, β) = m is the smallest integer m such that there is a chain of irreducible characters of H such that α = ψ 0 ∼ ψ 1 . . . ∼ ψ m = β. If there is no such chain then d(α, β) = −∞ and if α = β then the distance is zero. If X is the set of irreducible constituents of χ H then m(χ) := max α∈Irr(H) min ψ∈X d(α, ψ). The following results from [2] will be useful.

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“…The notion of combinatorial depth was defined in [1]. The related concept of ordinary depth has its origins in von-Neumann algebras, see [9].…”

Section: The Combinatorial Depth and Ordinary Depthmentioning

confidence: 99%

“…Let us denote by H x = x −1 Hx and H x 1 x n = H ∩ H x 1 ∩ · · · ∩ H x n , for x x 1 x n ∈ G. Let i = i H G = H x 1 x i x 1 x i ∈ G and = H G = ∪ i≥0 i , where 0 = H . In [1] the following characterization of combinatorial depth, d c H G is proved.…”

Section: The Combinatorial Depth and Ordinary Depthmentioning

confidence: 99%

The Depth of Subgroups of Suzuki Groups

Héthelyi

Horváth

Petényi

2015

Communications in Algebra

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“…Set It was shown recently in [1] that for any inclusion of finite groups H ⊂ G the inclusion kH ⊂ kG is always of finite depth for any field k. It is also not difficult to notice that any extension of semisimple Hopf algebras is of finite depth. It is however still an open question if an arbitrary extension of finite dimensional Hopf algebras has finite depth.…”

Section: Preliminariesmentioning

confidence: 99%

“…In the first section we recall the algebraic notion of depth for extensions of semisimple algebras from [1]. It extends the depth notion introduced in [5].…”

Section: Introductionmentioning

confidence: 99%

Subgroups of odd depth—a necessary condition

Burciu

2013

Czech Math J

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“…Following [1] an extension of k-algebras B ⊂ A is said to have left depth two (resp. right depth two) if there exists a positive integer n such that…”

Section: Introductionmentioning

confidence: 99%