I am studying $AC$-groups, i.e. groups in which the centralizer of every non-identity element is abelian. Now I need an example of a group in which the centralizer of every non-identity element is non-abelian. Where can I find and how can I classify them?
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5$\begingroup$ This was posted earlier and also answered on math.stackexchange.com/questions/1500269 so I don't know why you asked it again here. $\endgroup$– Derek HoltCommented Oct 27, 2015 at 22:53
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$\begingroup$ @Derek Holt : thanks but this question is different.Here i ask how can i classify them . $\endgroup$– Reza SheikhiCommented Oct 28, 2015 at 9:57
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1 Answer
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The smallest groups in which the centralizer of every element is non-abelian have order $32$. You can find them with GAP as follows:
gap> IsExample := G -> ForAll(ConjugacyClasses(G),
> cl->not IsAbelian(Centralizer(G,Representative(cl))));;
gap> n := 1;;
gap> repeat n := n + 1; sol := Filtered(AllGroups(n),IsExample); until sol <> [];
gap> sol;
[ <pc group of size 32 with 5 generators>, <pc group of size 32 with 5 generators> ]
gap> List(sol,IdGroup);
[ [ 32, 49 ], [ 32, 50 ] ]
gap> List(sol,StructureDescription);
[ "(C2 x D8) : C2", "(C2 x Q8) : C2" ]
answered Oct 27, 2015 at 22:36