Difference between revisions of "General linear group"
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The group of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436801.png" /> invertible matrices over an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436802.png" /> with a unit; the usual symbols are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436803.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436804.png" />. The general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436805.png" /> can also be defined as the automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436806.png" /> of the free right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436807.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436808.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436809.png" /> generators. | The group of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436801.png" /> invertible matrices over an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436802.png" /> with a unit; the usual symbols are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436803.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436804.png" />. The general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436805.png" /> can also be defined as the automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436806.png" /> of the free right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436807.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436808.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436809.png" /> generators. | ||
Revision as of 14:11, 29 January 2012
The group of all invertible matrices over an associative ring (cf. Associative rings and algebras)
with a unit; the usual symbols are
or
. The general linear group
can also be defined as the automorphism group
of the free right
-module
with
generators.
In research on the group its normal structure is of considerable interest. The centre
of the group
consists of scalar matrices with entries from the centre of the ring (cf. Centre of a ring)
. When
is commutative one defines the special linear group
, which consists of matrices with determinant 1. When
is a field, the commutator subgroup of the group
coincides with
(apart from the case
,
), and any normal subgroup of
is either contained in
or contains
. In particular, the projective special linear group
![]() |
is a simple group (apart from the cases ,
).
If is a skew-field and
, any normal subgroup of
is either contained in
or contains the commutator subgroup
of
generated by transvections (cf. Transvection), and the quotient group
is simple. Also, there exists a natural isomorphism
![]() |
where is the multiplicative group of the skew-field
. If
is finite-dimensional over its centre
, then the role of
is played by the group of all matrices from
with reduced norm 1. The groups
and
do not always coincide, although this is so if
is a global field (see Kneser–Tits hypothesis).
The study of the normal structure of general linear groups over a ring is associated with algebraic
-theory. The group
over a general ring
may contain numerous normal subgroups. For example, if
is a commutative ring without zero divisors and with a finite number of generators, then
is a residually-finite group, i.e. for each element
there exists a normal subgroup
of finite index not containing
. In the case
, the description of the normal subgroups of
is in fact equivalent to the congruence problem for
, since
![]() |
and any non-scalar normal subgroup of the group for
is a congruence subgroup.
There is a deep analogy between the structure of general linear groups and that of other classical groups. This analogy extends also to simple algebraic groups and Lie groups.
References
[1] | E. Artin, "Geometric algebra" , Interscience (1957) |
[2] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) |
[3] | H. Bass, "Algebraic ![]() |
General linear group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=General_linear_group&oldid=13698