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.

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, 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, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 2

Offset

1,16

Comments

The intersection of the normalizers of all subgroups of G is also called the Baer norm.

N(G) = Z(G) for every group of cubefree order. (See the Miles Englezou link for a proof.)

References

R. Baer, Norm and hypernorm, Publ. Math. Debrecen, 4 (1956), 347-350.

Links

Table of n, a(n) for n=1..88.

Wikipedia, Norm (group)

Miles Englezou, Proof that N(G) = Z(G) for groups of cubefree order

Formula

|N(G)| >= |Z(G)|. If n is a term of A051532 then a(n) = 0, since G = Z(G) = N(G).

By Baer (1956), Z(G) = 1 implies N(G) = 1. Hence no centerless group G satisfies |N(G)| <> |Z(G)|.

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.

Example

a(3) = 0 since Z(C3) = N(C3) = C3, and C3 is the only group of order 3.
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)|.

Prog

(GAP)
U:=[];; LoadPackage("sonata");;
for n in [1..64] do
    T:=[];;
    for i in [1..NrSmallGroups(n)] do
        S:=[];;
        G:=SmallGroup(n, i);;
        for k in [1..Length(Subgroups(G))] do
            S:=Concatenation(S, [Normaliser(G, Subgroups(G)[k])]);
        od;
        if Size(Intersection(S))<>Order(Centre(G)) then
            T:=Concatenation(T, [i]);
        fi;
    od;
    U:=Concatenation(U, [Size(T)]);
od; 	
Print(U);

Crossrefs

Cf. A051532, A046099, A004709.

Sequence in context: A308229 A318673 A151795 * A008437 A366076 A324326

Adjacent sequences: A375196 A375197 A375198 * A375200 A375201 A375202

Keyword

nonn

Author

Miles Englezou, Aug 11 2024

Status

approved