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A380147 Number of isoclinism classes of groups of order n.

%I A380147 #28 Jan 14 2025 09:54:53
%S A380147 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,
%T A380147 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,
%U A380147 3,1,4,1,20,1,2,2,2,1,6,1,11,3,2,1,9,1,2,1,4,1,8
%N A380147 Number of isoclinism classes of groups of order n.
%C A380147 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.
%C A380147 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].
%C A380147 A diagram of the mappings:
%C A380147                 fxf
%C A380147 Inn(G)xInn(G) ------> Inn(H)xInn(H)
%C A380147       |                     |
%C A380147    w1 |                     | w2
%C A380147       |                     |
%C A380147       \/                   \/
%C A380147     [G,G]    -------->    [H,H]
%C A380147                  g
%C A380147 If the diagram commutes, then G and H are isoclinic.
%H A380147 Miles Englezou, <a href="/A380147/a380147_5.txt">GAP Program</a>
%H A380147 The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Isoclinism_of_groups">Isoclinism of groups</a>
%H A380147 Wikipedia, <a href="https://en.wikipedia.org/wiki/Isoclinism_of_groups">Isoclinism of groups</a>
%F A380147 a(A051532(n)) = 1.
%e A380147 a(4) = 1 since both groups of order 4 are abelian and therefore form a single isoclinism class.
%e A380147 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.
%o A380147 (GAP) # See Miles Englezou link.
%Y A380147 Cf. A318895, A051532.
%Y A380147 A241276 is a lower bound.
%K A380147 nonn
%O A380147 1,6
%A A380147 _Miles Englezou_, Jan 13 2025