Isoclinism
Two groups G and H are isoclinic when there are isomorphisms
α : G/Z(G) → H/Z(H)β : G′ → H′
that agree on commutators: β([x, y]) = [x′, y′] whenever α(xZ) = x′Z and
α(yZ) = y′Z.
Isomorphic groups are isoclinic. The converse fails, and fails widely — the relation is about commutator structure alone, so it cannot see the abelian part a group carries alongside it.
A class spans orders
For any abelian A, G × A is isoclinic to G: the extra factor lands in the
center, so (G × A)/Z(G × A) ≅ G/Z(G) and (G × A)′ = G′. Every group is
therefore isoclinic to infinitely many larger ones.
D₄(order 8) is isoclinic toD₄ × C₂(16),D₄ × C₂²(32), …D₄andQ₈are isoclinic to each other — same commutator structure, and not isomorphic.
Abelian groups form a single class, the class of the trivial group: G′ = 1
and G/Z(G) = 1 say the same thing about all of them at once.
Stem groups
Every order in a class is a multiple of the class's least order, and the members
attaining it are the stem groups — those with Z(G) ≤ G′. Every class has
at least one, and a class may hold several non-isomorphic ones: D₄ and Q₈
are both stem in theirs.
A stem group is the smallest carrier of its class's commutator structure, so a
class is understood by understanding its stem groups: every member of the class
has the same G/Z(G) and the same G′ as they do.
Most members are a stem group with abelian padding, but not all of them.
M16 = ⟨a, b | a⁸ = b² = 1, aᵇ = a⁵⟩ is isoclinic to D₄ and Q₈, and it is
not S × A for any stem group S and non-trivial abelian A: it is directly
indecomposable. Its center is ⟨a²⟩ ≅ C₄ while D₄ × C₂ has center C₂², so
the two are different groups of order 16 in one class.
What it is used for
Isoclinism sorts groups by commutator behavior rather than by size, which makes
it the natural coarsening for questions that ignore central direct factors —
character degrees away from the center, the structure of G′, and the
classification of p-groups, where the classes of order p⁵ and p⁶ are a
standard organizing tool.