finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information...

theory. Sylow processed and further developed the innovative works of mathematicians Niels Henrik Abel and Évariste Galois in algebra. Sylow theorems and...

Sylow (1792–1875), Norwegian Minister of the Army Sylow theorems, a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow...

groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes...

(mod p). This is Wilson's theorem. It is possible to deduce Wilson's theorem from a particular application of the Sylow theorems. Let p be a prime. It is...

number of its elements) is a power of p. Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pn for every prime...

another proof of Burnside's theorem, because Burnside's theorem is used to prove this converse. A Sylow system is a set of Sylow p-subgroups Sp for each prime...

This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures...

group Product of group subsets Schur multiplier Semidirect product Sylow theorems Hall subgroup Wreath product Butterfly lemma Center of a group Centralizer...

the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done by the Gorenstein–Walter theorem which...

ISBN 978-0-486-81690-6. For the Sylow theorems see p. 43; for Lagrange's theorem, see p. 12; for Burnside's theorem see p. 143. Bryant, John; Sangwin...

not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order...

Theorems about abelian groups (i.e. modules over the principal ideal domain Z {\displaystyle \mathbb {Z} } ) can often be generalized to theorems about...

The Sylow subgroups of the symmetric groups are important examples of p-groups. They are more easily described in special cases first: The Sylow p-subgroups...

phrased simply as "normalizers grow". Every Sylow subgroup of G is normal. G is the direct product of its Sylow subgroups. If d divides the order of G, then...

elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct...

cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called...

conjugate Sylow theorem (for n odd): for n odd, each reflection, together with the identity, form a subgroup of order 2, which is a Sylow 2-subgroup...

prime and let n ≥ 1 {\displaystyle n\geq 1} . Let P {\displaystyle P} be a Sylow p-subgroup of the symmetric group S p n {\displaystyle S_{p^{n}}} . Then...

which has qd conjugates, for some d > 0. By the first statement of Sylow's theorem, G has a subgroup S of order pa. Because S is a nontrivial p-group...

G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of M ⁡ ( G ) {\displaystyle...

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

size of the 2-Sylow subgroup of SL2(F) is 2n, where n = ord2(p2 − 1) + ord2(r). The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups...

algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in (Higman...

{\displaystyle m\times m} upper triangular matrix. Any finite group whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular...

group generated by the element in Cauchy's theorem. Cauchy's theorem is generalized by Sylow's first theorem, which implies that if pn is the maximal power...

spacetime dimensions) associated with the Poincaré symmetry, by Noether's theorem, imply 10 conservation laws: 1 for the energy – associated with translations...

} with a2 + b2 = 1. This results from the spectral theorem by regrouping eigenvalues that are complex conjugate, and taking into account...

Sylow theorems Transcendence of e and π (as corollaries of Lindemann–Weierstrass) Tychonoff's theorem (to do) Ultrafilter lemma Ultraparallel theorem...

example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed...