OFFSET
1,6
COMMENTS
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.
LINKS
Miles Englezou, GAP Program
The Group Properties Wiki, Isoclinism of groups
Wikipedia, Isoclinism of groups
FORMULA
a(A051532(n)) = 1.
EXAMPLE
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.
PROG
(GAP) # See Miles Englezou link.
CROSSREFS
KEYWORD
nonn
AUTHOR
Miles Englezou, Jan 13 2025
STATUS
approved