报告题目：Injectors with a central socle in finitesoluble groups and extraspecially irreducible groups

报告时间：2015年7月15日9:30-11:30

　　　　　2015年7月17日9:30-11:30

报告地点：科学会堂A710

报告摘要：I.We report about some questions and recent results in the theory of Fitting classes of soluble groups. For a set of primesπ, defineZ^πto be the Fitting class of finite soluble groups whoseπ-socle is central. In response to an Open Question of Doerkand Hawkes [Finite Soluble Groups, de Gruyter, 1992; IXx4, p. 628], we have described different constructions for theZ^π-injectors of a finite soluble groups, which specialize to the characterization obtained by Frantz and Lockettwhenπ={p}. Also, another Open Question inthat referred book (IX§3, p. 615), first raised by Lockett,wonder whether the Fitting classZ^pis permutable. As stated in thementioned reference:“A negative answer here would settle the question whether there exist dominant Fitting classes which are not permutable”.However,we have recently obtained that in fact the Fitting classZ_is permutable, for any set of primesπ. So, the last mentioned question whether dominant Fitting classes are permutable remains open.

II..We present recent new constructions of some extraspecially irreducible groups, and discuss their uniqueness and automorhisms. These constructions have evolved to describe theminimal counterexample in the proof of the permutability of the Fitting classZ^πof finite soluble groups whoseπ-socle is central (πa set of primes). However, they appear to be general constructions, which can be of interest in many other different problems and contexts.