A051532 -id:A051532 - cp's OEIS frontend

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

Miles Englezou, Aug 11 2024

nonn

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.)

			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)|.

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

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

Cf. A051532, A046099, A004709.

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);

|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.

A380147 Number of isoclinism classes of groups of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 2, 2, 1, 7, 1, 2, 2, 2, 1, 4, 1, 8, 1, 2, 1, 7, 1, 2, 2, 5, 1, 6, 1, 2, 1, 2, 1, 14, 1, 4, 1, 3, 1, 11, 2, 5, 2, 2, 1, 9, 1, 2, 2, 27, 1, 4, 1, 3, 1, 4, 1, 20, 1, 2, 2, 2, 1, 6, 1, 11, 3, 2, 1, 9, 1, 2, 1, 4, 1, 8

Offset: 1

Miles Englezou, Jan 13 2025

nonn

Isoclinism is an equivalence relation on groups which generalizes isomorphism: it partitions nonisomorphic groups of the same order into classes. For example, all abelian groups of order k are isoclinic, and therefore belong to a single isoclinism class.
Two groups G and H are isoclinic if: there exists an isomorphism f between the inner automorphism groups Inn(G) and Inn(H); there exists an isomorphism g between the commutator subgroups [G,G] and [H,H]; and if f and g commute with the commutator maps w1:Inn(G)xInn(G) -> [G,G] and w2:Inn(H)xInn(H) -> [H,H].
A diagram of the mappings:
                fxf
Inn(G)xInn(G) ------> Inn(H)xInn(H)
      |                     |
   w1 |                     | w2
      |                     |
      \/                   \/
    [G,G]    -------->    [H,H]
                 g
If the diagram commutes, then G and H are isoclinic.

			a(4) = 1 since both groups of order 4 are abelian and therefore form a single isoclinism class.
a(8) = 2 since of the 5 groups of order 8, 3 are abelian and form a single isoclinism class, and the remaining 2 are isoclinic to each other. Therefore there are 2 isoclinism classes of order 8.

Miles Englezou, GAP Program
The Group Properties Wiki, Isoclinism of groups
Wikipedia, Isoclinism of groups

Cf. A318895, A051532.

A241276 is a lower bound.

GAP
```
# See Miles Englezou link.
```

a(A051532(n)) = 1.

A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

Original entry on oeis.org

1, 6, 12, 16, 24, 32, 48, 64, 96, 128, 256, 512, 1024, 2048

Offset: 1

Ben Branman, Feb 29 2012

			For a(n)=12, there are 2 Abelian groups and 3 nonabelian groups, so the ratio A000001(12)/A000688(12)=5/2=2.5, which beats the previous record of 2, so 12 is in the sequence.

H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
H. U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
H. U. Besche and B. Eick, The Groups of Order at Most 1000 Except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.
Eric Weisstein's World of Mathematics, Finite Group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Index entries for sequences related to groups

Cf. A000001, A000688, A060689, A051532.

s = {1}; a = 1; Do[b = FiniteGroupCount[n]/FiniteAbelianGroupCount[n];
  If[b > a, a = b; AppendTo[s, n]], {n, 1, 2047}]; s

a(14) from Eric M. Schmidt, Aug 02 2012

A335419 Integers m such that every group of order m is not simple.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Bernard Schott, Jul 09 2020

nonn

Officially, the group of order 1 is not considered to be simple; "a group <> 1 is simple if it has no normal subgroups other than G and 1" (See reference for Joseph J. Rotman's definition).
There is no prime term because there exists only one group of order p and this cyclic group Z/pZ is simple.
As a consequence of Feit-Thompson theorem, all odd composites are terms of this sequence.
The first composite even number that is not present in the data is 60 that is the order of simple alternating group Alt(5), the second one that is missing is 168 corresponding to simple Lie group PSL(3,2) [A031963].

			There exist 5 (nonisomorphic) groups of order 8: Z/8Z, Z/2Z × Z/4Z, (Z/2Z)^3, D_4 and H_8; none of these 5 groups is simple, so 8 is a term.
There exist 13 (nonisomorphic) groups of order 60 (see A000001), 12 are not simple but the alternating group Alt(5) is simple, hence 60 is not a term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, Exercice 1.44 p.96.
Joseph J. Rotman, The Theory of Groups: An Introduction, 4th ed., Springer-Verlag, New-York, 1995. Page 39, Definition.

Craig Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
Wikipedia, Feit-Thompson theorem.

Complement of A005180 (except for 1).
Subsequence: A014076 (odd nonprimes).
Cf. A000001, A031963, A051532 (similar for Abelian), A056867 (similar for nilpotent).

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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.

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A380147 Number of isoclinism classes of groups of order n.

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A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

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A335419 Integers m such that every group of order m is not simple.

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