the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup...

Cartan's theorem may refer to several mathematical results by Élie Cartan: Closed-subgroup theorem, 1930, that any closed subgroup of a Lie group is a...

if S ⊆ CG(T). For a subgroup H of group G, the N/C theorem states that the factor group NG(H)/CG(H) is isomorphic to a subgroup of Aut(H), the group...

Any topologically closed subgroup of a Lie group is a Lie group. This is known as the closed subgroup theorem or Cartan's theorem. The quotient of a...

a neighborhood of the identity in the SO(3). For a proof, see Closed subgroup theorem. The exponential map is surjective. This follows from the fact...

group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts...

corresponding subgroup is Aut(E/K), that is, the set of those automorphisms in Gal(E/F) which fix every element of K. The fundamental theorem says that this...

radicals (Abel–Ruffini theorem). This property is also used in complexity theory in the proof of Barrington's theorem. Consider the subgroups B = { [ ∗ ∗ 0 ∗...

∩ cl N is closed, then H is closed. Every discrete subgroup of a Hausdorff commutative topological group is closed. The isomorphism theorems from ordinary...

conjugacy-closed subgroup of the general linear group over the extension field. This result is typically referred to as a stability theorem. A subgroup is said...

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every...

Chevalley's structure theorem on algebraic groups: if G is an algebraic group then it contains a unique closed normal subgroup N such that N is affine...

the closure of the subgroup <P> generated by P will be finite, or some torus T′ contained in T. The original Kronecker's theorem (Leopold Kronecker,...

In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group...

intended meaning (and in fact maximal proper subgroups are not in general compact). The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group...

coordinates, exponential coordinates or normal coordinates. See the closed-subgroup theorem for an example of how they are used in applications. Remark: The...

the proof of the Feit–Thompson odd-order theorem. Each maximal subgroup M has a certain nilpotent Hall subgroup Mσ with normalizer contained in M, whose...

theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G. If the field K is not algebraically closed, the theorem can...

constructions below. Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem. Then the Lie algebra of G may be...

realised as a finite, hence Zariski-closed, subgroup of some G L n {\displaystyle \mathrm {GL} _{n}} by Cayley's theorem). In addition it is both affine and...

its kernel is exactly the subgroup Homeo 0 ⁡ ( S ) {\displaystyle \operatorname {Homeo} _{0}(S)} . The Dehn–Nielsen–Baer theorem states that it is in addition...

the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and Øystein Ore. Both Dirac's and Ore's theorems can also be derived...

commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...

isomorphic to f(G), the image of G under f (which is a subgroup of H also), by the first isomorphism theorem for groups. Let R and S be rings (assumed unital)...

groups, and combinatorics. Cayley's theorem states that every group G {\displaystyle G} is isomorphic to a subgroup of the symmetric group on (the underlying...

the Borel subgroup of G L ( n ) {\displaystyle GL(n)} . It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of G L ( n...

extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem. Von Neumann was the first to axiomatically define an abstract Hilbert...

theorem (geometric group theory) Focal subgroup theorem (abstract algebra) Frobenius determinant theorem (group theory) Frobenius reciprocity theorem...

In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can...

In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner...