Isoclinic groups

Definition
Two groups are said to be isoclinic if there is an defining ingredient::isoclinism between them, i.e., there is an isomorphism between their inner automorphism groups as well as an isomorphism between their derived subgroups such that the isomorphisms are compatible with the commutator map $$\operatorname{Inn}(G) \times \operatorname{Inn}(G) \to G'$$.

Invariants under isoclinism
Many arithmetic functions associated with groups are invariant under isoclinism, and many group properties are preserved under isoclinism. Some of these are listed below:

Groups it is made out of

 * Isoclinic groups have same non-abelian composition factors

Probabilistic invariants

 * Isoclinic groups have same commuting fraction

Stem groups
A stem group is a group whose center is contained in its derived subgroup. The following are true:


 * Every group is isoclinic to a stem group
 * The unique stem group among abelian groups is the trivial group
 * Stem group has the minimum order among all groups isoclinic to it. In fact, the order of a stem group must divide the order of any group isoclinic to it. In particular, isoclinic stem groups have the same order (though they need not be isomorphic).

More on prime powers
The classification of groups of order $$2^n, n \le 6$$ by Hall and Senior was done on the basis of isoclinism. In the jargon used by Hall and Senior, they defined the Hall-Senior family of a group as its equivalence class under isoclinism, and the Hall-Senior genus (see Hall-Senior genus) was obtained by further refinement based on the lattice of normal subgroups and the Hall-Senior family of each normal subgroup. To see this classification in action, refer:


 * Groups of order 8
 * Groups of order 16
 * Groups of order 32
 * Groups of order 64

Related group and subgroup properties

 * Subgroup isoclinic to the whole group
 * Group having no proper isoclinic subgroup
 * Finite group that is not isoclinic to a group of smaller order

Original use

 * : Definition introduced on Page 133 (Page 4 within the paper)