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 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:

Simple invariants

Arithmetic function Group property for which this is defined (this also forces that if one group has the property, so does any isoclinic group) Meaning Proof of invariance under isoclinisms Value for abelian groups and the trivial group exception
nilpotency class nilpotent group Length of the upper central series or the lower central series, or any central series of minimum possible length Isoclinic groups have same nilpotency class 0 or 1. Note that the only situation where isoclinic groups can have different nilpotency class is the case of the trivial group (class zero) and a nontrivial abelian group (class one).
derived length solvable group Length of the derived series Isoclinic groups have same derived length 0 or 1. Note that the only situation where isoclinic groups can have different derived length is the case of the trivial group (length zero) and a nontrivial abelian group (length one).

Multiset invariants

Multiset invariant Proof of something close to invariance under isoclinisms Full statement Explanation
degrees of irreducible representations isoclinic groups have same proportions of degrees of irreducible representations Suppose $G_1$ and $G_2$ are finite groups that are isoclinic and $d$ is a positive integer. Then, if $m_1$ is the number of irreducible representations (up to equivalence) of $G_1$ of degree $d$ over $\mathbb{C}$, and $m_2$ is the number of irreducible representations (up to equivalence) of $G_2$ of degree $d$ over $\mathbb{C}$, then $m_1/m_2 = |G_1|/|G_2|$. Note in particular that this implies that the set of degrees of irreducible representations is the same for both groups. The idea is that an irreducible projective representation of the inner automorphism group lifts to $G_1$ if and only if it lifts to $G_2$, and the ratio of the number of lifts is proportional to the order. For more, see the full proof.
conjugacy class sizes isoclinic groups have same proportions of conjugacy class sizes Suppose $G_1$ and $G_2$ are finite groups that are isoclinic and $d$ is a positive integer. Then, if $m_1$ is the number of irreducible representations (up to equivalence) of $G_1$ of degree $d$ over $\mathbb{C}$, and $m_2$ is the number of irreducible representations (up to equivalence) of $G_2$ of degree $d$ over $\mathbb{C}$, then $m_1/m_2 = |G_1|/|G_2|$. Note in particular that this implies that the set of conjugacy class sizes is the same for both groups. The idea is that for each element in the inner automorphism group, the size of the conjugacy class of any element of $G_1$ mapping to it is the same as the size of the conjugacy class of any element of $G_2$ mapping to it.

Facts

Taking the closure of group properties under isoclinism

Starting group or group property Meaning Property of being isoclinic to a group with this property Meaning
trivial group only one element abelian group any two elements commute
finite group finitely many elements FZ-group the center has finite index. Note that the derived subgroup is forced to be finite because FZ implies finite derived subgroup.

Stem groups

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

Isoclinism for small orders

Order Total number of groups up to isomorphism Number of equivalence classes under isoclinism among the groups of that order List of groups for each equivalence class under isoclinism Total number of stem groups Number of equivalence classes under isoclinism for stem groups of that order List of groups for each equivalence class under isoclinism
1 1 1 trivial group (stem group) 1 1 trivial group
2 1 1 cyclic group:Z2 (stem group: trivial group) 0 0 --
3 1 1 cyclic group:Z3 (stem group: trivial group) 0 0 --
4 2 1 class of (cyclic group:Z4 and Klein four-group) (stem group: trivial group) 0 0 --
5 1 1 cyclic group:Z5 (stem group: trivial group) 0 0 --
6 2 2 class of cyclic group:Z6 (stem group: trivial group)
other class contains symmetric group:S3 (stem group)
1 1 symmetric group:S3
7 1 1 cyclic group:Z7 (stem group: trivial group) 0 0 --
8 5 2 class of (cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8) (stem group: trivial group)
class of (dihedral group:D8 and quaternion group)
2 1 class of (dihedral group:D8 and quaternion group) -- see dihedral and dicyclic groups are isoclinic
9 2 1 class of (cyclic group:Z9 and elementary abelian group:E9) (stem group: trivial group) 0 0 --
10 2 2 class of cyclic group:Z10 (stem group: trivial group)
class of dihedral group:D10
1 1 dihedral group:D10
12 5 3 class of (cyclic group:Z12 and direct product of Z6 and Z2) (stem group: trivial group)
class of (dicyclic group:Dic12 and dihedral group:D12) (stem group: symmetric group:S3)
class of alternating group:A4
1 1 alternating group:A4
16 14 3 class of abelian groups (5 members)
class of class two groups (6 members)
class of class three groups (3 members)
see Groups of order 16#Families and classification for more.
3 1 class three groups: dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16

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: