Isoclinic groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Two groups are said to be isoclinic if there is an isoclinism between them, i.e., there are isomorphisms between their inner automorphism groups as well as isomorphisms between their derived subgroups such that the isomorphisms are compatible with the commutator map $\mathrm{I}\mathrm{n}\mathrm{n}(G)\times \mathrm{I}\mathrm{n}\mathrm{n}(G)\to G^{\prime }$.

Facts

Groups isoclinic to the trivial group

A group is isoclinic to the trivial group if and ony if it is abelian. In that case, the inner automorphism group and derived subgroup are both trivial, and thus the isomorphisms are just the trivial maps.

Subgroups isoclinic to each other

Any subgroup of a group is isoclinic to its product with the center of the group. In particular, this means that any two subgroups having nonempty intersection with the same cosets of the center of the whole group are isoclinic.

In particular, any cocentral subgroup of a group is isoclinic to the whole group.

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:

Simple invariants

• Isoclinic groups have same nilpotency class (with the minor issue of class zero versus one; in particular, any group isoclinic to a nilpotent group is nilpotent).
• Isoclinic groups have same derived length (in particular, any group isoclinic to a solvable group is solvable)
• Isoclinic groups have same non-abelian composition factors

Multiset invariants

• Isoclinic groups have same proportions of degrees of irreducible representations: The degrees of irreducible representations are the same, with the number of occurrences of each degree scaled in proportion to the order of the group. In particular, isoclinic groups of the same order have precisely the same degrees of irreducible representations.
• Isoclinic groups have same proportions of conjugacy class sizes: The conjugacy class sizes are the same, with the number of occurrences of each conjugacy class size scaled in proportion to the order of the group.

Probabilistic invariants

• Isoclinic groups have same commuting fraction

Use of isoclinism in classification

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#Families and classification
• Groups of order 16#Families and classification
• Groups of order 32#Families and classification
• Groups of order 64#Families and classification

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