0

I'm a complete beginner in finite groups, and I'm doing a related project with my classmates in this semester. So I have a certainly concrete problem in a computational fact of group theory: I need to find all the groups of order <2001 with a composition factor $A_5$. I don't wanna to directly use magma (programming) to find the answer with brute force. Can any veteran tell me whether there's any other valuable approach to tackle this problem? or we must program to solve it? thanks!

youknowwho
  • 1,407
  • 2
    Is there a connection with your very recent question : https://math.stackexchange.com/q/3670992 ? – Jean Marie May 12 '20 at 11:25
  • oh yep, i'm sorry. I'm not very familiar with rules here. Maybe I should merge them. – youknowwho May 12 '20 at 11:51
  • 2
    To be brutally frank, if you are really a complete beginner in finite groups, then I don't think you have any chance whatsoever of (for example) classifying the groups of order $1920 = 2^5|A_5|$ that have $A_5$ as a composition factor. That is not at all easy and requires knowledge of the representation theory of $A_5$ and $S_5$ over the field of order $2$, calculating cohomology groups, etc. You would do better to find the groups using the databases in GAP or Magma, and then studying their properties. – Derek Holt May 12 '20 at 12:36
  • I just checked in Magma and there are 588 isomorphism classes of groups of order 1920 having $A_5$ as a composition factor. – Derek Holt May 12 '20 at 12:45
  • @DerekHolt Thank u so much; I once thought there might be any elementary way to think about this problem, and now I understand my situation ^_^. – youknowwho May 12 '20 at 13:02
  • @DerekHolt Forgive me to ask a further technical question: is their any code/command in magma that can record appearance of a certain group in a composition series? – youknowwho May 12 '20 at 13:13
  • $\mathsf{name := CompositionFactors(Alt(5))[1])}$; will give you Magma's name for $A_5$, and then you can test its presence as a composition factor of group $\mathsf{G}$ with $\mathsf{name\ in\ CompositionFactors(G);}$. Also, you can quickly find all insoluble groups of a given order with $\mathsf{gps := SmallGroups(1920 : Search:="Insoluble");}$. – Derek Holt May 12 '20 at 13:42

0 Answers0