A question from the Kourovka notebook on formation products

Abstract

It is shown in this paper that if X is a class of simple groups such that TT(JE) = char X, the 3E-saturated formation f) generated by a finite group cannot be expressed as the Gaschiitz product J? o <S of two non-3E-saturated formations if ¥j ^ <5. It answers some open questions on products of formations. The relation between cj-saturated and SE-saturated formations is also discussed.

On 𝔛-Saturated Formations of Finite Groups

“…But, in general, the family of X ω -saturated formations does not coincide with the one of ω-saturated formations, as it is shown in the following example (see [6] for more details):…”

“…It is shown in [6] that ω-saturated formations are X ω -saturated, where X ω is the class of all simple ω-groups. But, in general, the family of X ω -saturated formations does not coincide with the one of ω-saturated formations, as it is shown in the following example (see [6] for more details):…”

“…We have that G is a monolithic group whose socle is an ω-group. We know that F is X ω -saturated by [6;Section 3], where X ω is the class of all simple ω-groups. By Corollary 4.10, Lemma 4.12,it follows that N ω F is ω-saturated and, hence, G ∈ N ω F. Bearing in mind that O ω (G) = 1, we conclude G ∈ bform(F), a contradiction.…”

Products of formations of finite groups

“…In [16], it is shown that ω-saturated formations are X ω -local, where X ω is the class of all simple ω-groups. However, the converse does not hold.…”

Factorizations of one-generated % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgzgjxyRrxDYbqehuuDJXwAKbIrYf2A0vNC % aGGbaiab-Db8ybaa!4772! $$ \mathfrak{X} $$ -local formations

“…A common extension of Skiba's theorems was presented in [7] by using the concept of X-local formation, where X is a class of simple groups with a completeness property.…”