In mathematics, specifically

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, isoclinism is an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

on groups which generalizes

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

. Isoclinism was introduced by to help classify and understand

p-group In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integ ...

s, although it is applicable to all groups. Isoclinism also has consequences for the

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \op ...

and the associated aspects of

character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information a ...

, as described in and . The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope. Some textbooks discussing isoclinism include and and .

Definition

The isoclinism class of a group ''G'' is determined by the groups ''G''/''Z''(''G'') (the

inner automorphism group In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group i ...

) and ''G''′ (the

commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

) and the commutator map from ''G''/''Z''(''G'') × ''G''/''Z''(''G'') to ''G''′ (taking ''a'', ''b'' to ''aba⁻¹b⁻¹''). In other words, two groups ''G''₁ and ''G''₂ are isoclinic if there are isomorphisms from ''G''₁/''Z''(''G''₁) to ''G''₂/''Z''(''G''₂) and from ''G''₁′ to ''G''₂′ commuting with the commutator map.

Examples

All

Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

s 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 ''G''×''A'' if and only if ''A'' is abelian. The dihedral,

quasidihedral In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer ''n'' greater than or equal to 4, there are exactly four isomorphism classes of non ...

, and

quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a nonabelian group, non-abelian group (mathematics), group of Group order, order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. ...

s of order 2^''n'' are isoclinic for ''n''≥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 In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

(also called the commutator subgroup), . Some enumeration results on isoclinism families are given in . Isoclinism is used in theory of

projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL( ...

s 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 group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

s .

References

* * * * * * * {{DEFAULTSORT:Isoclinism Of Groups Finite groups