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- W2111496055 abstract "p ≥ t(G) ≥ e(p− 1) + 1, where pe is the order of a Sylow p-subgroup of G. H. Fukushima [1] characterized a group G of p-length 2 satisfying t(G) = e(p− 1) + 1, see also [4]. Unfortunately, his characterization holds under a condition such that the p′-part V = Op′,p(G)/Op(G) of G is abelian. In this paper, using Dickson near fields, we shall give an explicit example (see Example 1) such that a group G of p-length 2 has the non abelian p′part V and satisfies t(G) = e(p− 1)+1. This example will be new and have a contributions in our research. Example 2 is also very interesting because quite different objects (see [3] and [5]) are unified on the ground of Dickson near fields. Let H be a sharply 2-fold transitive group on ∆ = {0, 1, α, β, . . . , γ} (see [8, p. 22]). Let V = H0 be a stabilizer of 0, and let U be the set consisting of the identity e and fixed point-free permutations in H. Then U is an elementary abelian p-subgroup of H with the order ps (see Lemma 1). Let σ be a permutation of order p on ∆ satisfying conditions σHσ−1 ⊆ H, σ = 1, σ(0) = 0 and σ(1) = 1. Then it is easy to see σUσ−1 ⊆ U and σV σ−1 ⊆ V . We set W = 〈σ〉 and CV (σ) = {v ∈ V | σv = vσ}. Assume that there exists a normal subgroup T of WV contained in V such that V is a semi-direct product of T by CV (σ). We set G = 〈W,T,U〉. Now, we present the well known results Lemmas 1 and 2 for completeness of this paper." @default.
- W2111496055 created "2016-06-24" @default.
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- W2111496055 date "2002-01-01" @default.
- W2111496055 modified "2023-09-26" @default.
- W2111496055 title "On the Nilpotency Index of the Radical of a Group Algebra. XI" @default.
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