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I am looking for the properties of groups having "immediate Descendants", in other therm, "Capable Groups"; The problem that I fond is that "Capable Group" could have many meaning!

So, could you please help me to optimize my "bibliographical" research?

I am using data bases like Mathscinet

Thank you very much

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    $\begingroup$ Just wikipedia: In mathematics, in the realm of group theory, a group is said to be capable if it occurs as the inner automorphism group of some group. Reference: R. Baer.See also the posts on this site, e.g. here. $\endgroup$ – Dietrich Burde Dec 7 '18 at 10:19
  • $\begingroup$ @DietrichBurde exactly Professor, also "if it is a central factor group", and "If it has immediate descendants"...; Are these equivalent definitions? They don´t seem so! $\endgroup$ – A.Messab Dec 7 '18 at 10:22
  • $\begingroup$ @DietrichBurde In fact the definition you gave me and "if it is a central factor group" can be seen easily as equivalent; The real problem is with "If it has immediate descendants"! $\endgroup$ – A.Messab Dec 7 '18 at 10:28
  • $\begingroup$ @DietrichBurde The idea is how to restrict my research on "capable groups of the definition "If it has immediate descendants"? $\endgroup$ – A.Messab Dec 7 '18 at 10:37
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    $\begingroup$ Yes, Hall has discussed these equivalent properties in his publication "The classification of prime-power groups" of 1940, page 137. A group is capable if it has immediate descendents. $\endgroup$ – Dietrich Burde Dec 7 '18 at 10:50

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