227-01227-01227-01-property and normal subgroups

On the converse of Hall's theorem

“…Ward [2] generalized the above result by showing that a group G is soluble if G ∈ E 2, 2 ′ ∩ E 3, 3 ′ . The complete classification of the class E π, π ′ was established by Z. Arad and E. Fisman in [1], and that of the class D π, π ′ was established by A. L. Gilotti in [8]. Here we quote A. L. Gilotti's result for the completeness (with a rearrangement).…”

“…In both cases, we have that D ∈ E 3 ′ because π = {3} and π ′ = {3}. Hence by [1,Corollary 5.6], D ∼ = P SL (2,8), a contradiction occurs. Thus G is π-separable if G ∈ V π, π ′ .…”

“…(see[8, Theorem 3.2].) A group G ∈ D π, π ′ if and only if for every composition factor D of G, one of the following holds:(1) either π(D) ⊆ π or π(D) ⊆ π ′ .…”

Carter and Gaschütz theories beyond soluble groups

“…On the other hand, by hypothesis and Remark 2.9, G satisfies D π and D π , and we may assume that the group is neither a π-group nor a π -group. In [13,Lemma 3.1], such a simple group is characterized to be G = P SL (2, q), where q > 3, q(q − 1) ≡ 0(3), q ≡ −1(4) and π(q + 1) ⊆ π, π(q(q − 1)/2) ⊆ π . Hence |π(G) ∩ π | ≥ |π(q(q − 1)/2)| ≥ 2, which is not possible and proves that G is π-separable.…”

“…Lemma 4. 13 Let H be a saturated formation, X be a group and H be an H-projector of X . Then H ∩ X H ≤ (X H ) .…”