In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated aspects of character theory, as described in and . The word "isoclinism" comes from the Greek ùÃÂÿúûùý÷àmeaning equal slope.
Some textbooks discussing isoclinism include and and .
The isoclinism class of a group G is determined by the groups G/Z(G) (the inner automorphism group) and ' (the commutator subgroup) and the commutator map from G/Z(G) àG/Z(G) to ' (taking a, b to aba<sup>âÂÂ1</sup>b<sup>âÂÂ1</sup>).
In other words, two groups G<sub>1</sub> and G<sub>2</sub> are isoclinic if there are isomorphisms from G<sub>1</sub>/Z(G<sub>1</sub>) to G<sub>2</sub>/Z(G<sub>2</sub>) and from G<sub>1</sub> to G<sub>2</sub> commuting with the commutator map.
All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and G is isoclinic with GÃÂA if and only if A is abelian. The dihedral, quasidihedral, and quaternion groups of order 2<sup>n</sup> are isoclinic for nâÂÂ¥3, in more detail.
Isoclinism divides p-groups into families, and the smallest members of each family are called stem groups. A group is a stem group if and only if Z(G) ⤠[G,G], that is, if and only if every element of the center of the group is contained in the derived subgroup (also called the commutator subgroup), . Some enumeration results on isoclinism families are given in .
Isoclinism is used in theory of projective representations of finite groups, as all Schur covering groups of a group are isoclinic, a fact already hinted at by Hall according to . This is used in describing the character tables of the finite simple groups .