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I know that $p$-groups have the property, and all nilpotent groups also do. I wanted to have some non-nilpotent examples. Then I checked quasisimple groups, and found every proper normal subgroups of a quasisimple is contained in the center.

Quasisimple group ${\rm SL}(2,q)$ would be an example I want, when $q>3$ is an odd. For an odd $q>3$, the center of ${\rm SL}(2,q)$ is of order 2 and that makes ${\rm SL}(2,q)$ a non-simple quasisimple group with the property held; otherwise, for an even $q>3$, ${\rm SL}(2,q)$ has trivial center. Although ${\rm SL}(2,3)$ is not quasisimple, ${\rm SL}(2,3)$ also holds the property and is hopefully the example of the smallest order. My question is: How to prove ${\rm SL}(2,3)$ is the example of the smallest order? What are other kinds of groups that may be examples?

Thank you, any help will be appreciated!

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  • $\begingroup$ $SL(2,3)$ is indeed the smallest non-nilpotent example, but I don't know of an easy way to prove this without considering many cases (or using a computer). $\endgroup$ – verret Apr 26 '20 at 4:41
  • $\begingroup$ @verret thanks. Is there any reference about that? $\endgroup$ – user517681 Apr 26 '20 at 4:44
  • $\begingroup$ No, I just checked in magma right now. $\endgroup$ – verret Apr 26 '20 at 4:44
  • $\begingroup$ @verret I also want to try it in magma, but I’m totally new. Can it be realized in the free web page version? If it can, could you please help me with the code? Thank you! $\endgroup$ – user517681 Apr 26 '20 at 5:15
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    $\begingroup$ You can for example run "IsIsomorphic(SL(2,3),SmallGroup(24,3));" or you can check people.maths.bris.ac.uk/~matyd/GroupNames $\endgroup$ – verret Apr 26 '20 at 20:26
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Here's a quick way to check that $\mathrm{SL}_2(3)$ has the desired property using the computer algebra system GAP:

grps:=AllSmallGroups([1..24],IsNilpotentGroup,false);; 
list:=Filtered(grps,x->Order(Center(x))>1);  

for g in list do
Print(StructureDescription(g),"  ",IsSubgroup(Center(g),Socle(g)),"\n");
od;

C3 : C4  false
D12  false
C3 x S3  false
C5 : C4  false
D20  false
C3 : C8  false
SL(2,3)  true
C3 : Q8  false
C4 x S3  false
D24  false
C2 x (C3 : C4)  false
(C6 x C2) : C2  false
C2 x A4  false
C2 x C2 x S3  false
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