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 relation ...
on
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 is ...
. 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 integer ...
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 \oper ...
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 about ...
, 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 its ...
) 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 commut ...
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, and
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
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 G ...
s of
finite 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, as all
Schur cover
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 \oper ...
ing 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