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A375199 Number of groups G of order n such that |N(G)| <> |Z(G)|, where N(G) is the intersection of the normalizers of all subgroups of G and Z(G) is the center of G.

%I A375199 #34 Sep 16 2024 09:23:02
%S A375199 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,11,0,0,
%T A375199 0,0,0,0,0,2,0,0,0,0,0,0,0,8,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,39,0,0,0,0,
%U A375199 0,0,0,5,0,0,0,0,0,0,0,10,3,0,0,0,0,0,0,2
%N A375199 Number of groups G of order n such that |N(G)| <> |Z(G)|, where N(G) is the intersection of the normalizers of all subgroups of G and Z(G) is the center of G.
%C A375199 The intersection of the normalizers of all subgroups of G is also called the Baer norm.
%C A375199 N(G) = Z(G) for every group of cubefree order. (See the Miles Englezou link for a proof.)
%D A375199 R. Baer, Norm and hypernorm, Publ. Math. Debrecen, 4 (1956), 347-350.
%H A375199 Wikipedia, <a href="https://en.wikipedia.org/wiki/Norm_(group)">Norm (group)</a>
%H A375199 Miles Englezou, <a href="/A375199/a375199.txt">Proof that N(G) = Z(G) for groups of cubefree order</a>
%F A375199 |N(G)| >= |Z(G)|. If n is a term of A051532 then a(n) = 0, since G = Z(G) = N(G).
%F A375199 By Baer (1956), Z(G) = 1 implies N(G) = 1. Hence no centerless group G satisfies |N(G)| <> |Z(G)|.
%F A375199 a(n) > 0 only when n is divisible by a cube (i.e., when n is a term of A046099). Equivalently, a(n) = 0 when n is a term of A004709.
%e A375199 a(3) = 0 since Z(C3) = N(C3) = C3, and C3 is the only group of order 3.
%e A375199 a(8) = 1 since |Z(Q8)| = 2 and |N(Q8)| = 8, and for other groups G of order 8 we get |N(G)| = |Z(G)|.
%o A375199 (GAP)
%o A375199 U:=[];; LoadPackage("sonata");;
%o A375199 for n in [1..64] do
%o A375199     T:=[];;
%o A375199     for i in [1..NrSmallGroups(n)] do
%o A375199         S:=[];;
%o A375199         G:=SmallGroup(n,i);;
%o A375199         for k in [1..Length(Subgroups(G))] do
%o A375199             S:=Concatenation(S,[Normaliser(G,Subgroups(G)[k])]);
%o A375199         od;
%o A375199         if Size(Intersection(S))<>Order(Centre(G)) then
%o A375199             T:=Concatenation(T,[i]);
%o A375199         fi;
%o A375199     od;
%o A375199     U:=Concatenation(U,[Size(T)]);
%o A375199 od;	
%o A375199 Print(U);
%Y A375199 Cf. A051532, A046099, A004709.
%K A375199 nonn
%O A375199 1,16
%A A375199 _Miles Englezou_, Aug 11 2024