In mathematical
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 ...
, the Schur multiplier or Schur multiplicator is the second
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of a group ''G''. It was introduced by in his work on
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.
Examples and properties
The Schur multiplier
of a finite group ''G'' is a finite
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 ...
whose
exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
divides the order of ''G''. If a
Sylow ''p''-subgroup of ''G'' is cyclic for some ''p'', then the order of
is not divisible by ''p''. In particular, if all
Sylow ''p''-subgroups of ''G'' are cyclic, then
is trivial.
For instance, the Schur multiplier of the
nonabelian group of order 6
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of Order of a group, order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest ...
is the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
since every Sylow subgroup is cyclic. The Schur multiplier of the
elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian group ...
of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the
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 ...
is trivial, but the Schur multiplier of
dihedral 2-groups has order 2.
The Schur multipliers of the finite
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
s are given at the
list of finite simple groups
A ''list'' is any set of items in a row. List or lists may also refer to:
People
* List (surname)
Organizations
* List College, an undergraduate division of the Jewish Theological Seminary of America
* SC Germania List, German rugby union ...
. The
covering groups of the alternating and symmetric groups In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classifie ...
are of considerable recent interest.
Relation to projective representations
Schur's original motivation for studying the multiplier was to classify
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 a group, and the modern formulation of his definition is the second
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
. A projective representation is much like a
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
except that instead of a homomorphism into the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, one takes a homomorphism into the
projective general linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
. In other words, a projective representation is a representation modulo the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
.
showed that every finite group ''G'' has associated to it at least one finite group ''C'', called a Schur cover, with the property that every projective representation of ''G'' can be lifted to an ordinary representation of ''C''. The Schur cover is also known as a covering group or Darstellungsgruppe. The Schur covers of the
finite simple groups
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 ...
are known, and each is an example of a
quasisimple group
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence
:1 \to Z(E) \to E \to S \to 1
such that E = , E/ ...
. The Schur cover of a
perfect group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universa ...
is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to
isoclinism In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism al ...
.
Relation to central extensions
The study of such covering groups led naturally to the study of
central
Central is an adjective usually referring to being in the center of some place or (mathematical) object.
Central may also refer to:
Directions and generalised locations
* Central Africa, a region in the centre of Africa continent, also known as ...
and stem extensions.
A
central extension of a group ''G'' is an extension
:
where
is a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of ''C''.
A stem extension of a group ''G'' is an extension
:
where
is a subgroup of the intersection of the center of ''C'' and 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 ...
of ''C''; this is more restrictive than central.
If the group ''G'' is finite and one considers only stem extensions, then there is a largest size for such a group ''C'', and for every ''C'' of that size the subgroup ''K'' is isomorphic to the Schur multiplier of ''G''. If the finite group ''G'' is moreover
perfect, then ''C'' is unique up to isomorphism and is itself perfect. Such ''C'' are often called universal perfect central extensions of ''G'', or covering group (as it is a discrete analog of the
universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
in topology). If the finite group ''G'' is not perfect, then its Schur covering groups (all such ''C'' of maximal order) are only
isoclinic In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism al ...
.
It is also called more briefly a universal central extension, but note that there is no largest central extension, as the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of ''G'' and an
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 ...
form a central extension of ''G'' of arbitrary size.
Stem extensions have the nice property that any lift of a generating set of ''G'' is a generating set of ''C''. If the group ''G'' is
presented in terms of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
''F'' on a set of generators, and a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
''R'' generated by a set of relations on the generators, so that
, then the covering group itself can be presented in terms of ''F'' but with a smaller normal subgroup ''S'', that is,
. Since the relations of ''G'' specify elements of ''K'' when considered as part of ''C'', one must have