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Theorems that help decompose a finite group based on prime factors of its order
finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information
Sylow_theorems
Surname list
Sylow is a surname that originates in Denmark. Notable people with the surname include: Peter Ludvig Sylow (1832–1918), Norwegian mathematician Ludvig
Sylow
Norwegian mathematician
Meidell Sylow (/ˈsyːlɔv/) (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory. Sylow processed
Peter_Ludvig_Sylow
Type of group in abstract algebra
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
Symmetric_group
Mathematical group based upon a finite number of elements
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
Finite_group
Topics referred to by the same term
Ludvig Sylow may refer to: Peter Ludvig Sylow (1832–1918), Norwegian mathematician Ludvig Sylow (DBU) (1861–1933), Danish football executive This disambiguation
Ludvig_Sylow
of primes in π, and whose index is not divisible by any primes in π. Any Sylow subgroup of a group is a Hall subgroup. The alternating group A4 of order
Hall_subgroup
Theorem classifying finite simple groups
Groups of 2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done
Classification of finite simple groups
Classification_of_finite_simple_groups
Football tournament
Sylow-Tournament (Danish: Sylow-Turneringen) was a knockout association football competition contested annually between 1918 and 1926, organised by the
Sylow-Tournament
Group of symmetries of a regular polygon
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
Dihedral_group
Danish football executive (1861–1933)
Ludvig Sylow (6 October 1861 – 20 February 1933) was a Danish football executive, who was the fifth chairman of the Danish Football Association from 1911
Ludvig_Sylow_(DBU)
Sporadic simple group
classes of maximal subgroups of J4 which are listed in the table below. A Sylow 3-subgroup of J4 is a Heisenberg group: order 27, non-abelian, all non-trivial
Janko_group_J4
Theorem on prime numbers
hand, each Sylow p-subgroup in S p {\displaystyle S_{p}} is a copy of C p {\displaystyle C_{p}} . Hence it follows that the number of Sylow p-subgroups
Wilson's_theorem
Topic in group theory
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
Wreath_product
Mathematical concept
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
Nilpotent_group
Commutative group (mathematics)
{\displaystyle G} it suffices to compute the automorphism groups of the Sylow p {\displaystyle p} -subgroups separately (that is, all direct sums of cyclic
Abelian_group
Name list
teacher Ludvig Strigeus, Swedish programmer Peter Ludvig Sylow, Norwegian mathematician Ludvig Sylow (DBU), Danish football executive This page or section
Ludvig
Group in which the order of every element is a power of p
(the 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
P-group
Norwegian politician
Arnoldus von Westen Sylow Koren (22 July 1764 – 8 October 1854) was a civil servant and district judge. He served as a representative at the Norwegian
Arnoldus von Westen Sylow Koren
Arnoldus_von_Westen_Sylow_Koren
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Monster_group
Class of mathematical groups
normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups
C-group
Aspect of mathematical group theory
these subgroups. Alternately, one could use the Sylow theorems, which state generally that all Sylow p-subgroups are conjugate. The projective linear
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Finite group
p-complement and any Sylow p-subgroup. A group is called p-nilpotent if it has a normal p-complement. Cayley showed that if the Sylow 2-subgroup of a group
Normal_p-complement
Quasisimple subnormal subgroup of a finite group
nilpotent) groups, p-component is used in a different sense to mean the Sylow p-subgroup, so the abelian group is the product of its p-components for
Component_(group_theory)
Walter (1965a, 1965b, 1965c), states that if a finite group G has a dihedral Sylow 2-subgroup, and O(G) is the maximal normal subgroup of odd order, then G/O(G)
Gorenstein–Walter_theorem
Finite group
groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups. The Sylow 2-subgroups of the following groups are quasidihedral: PSL3(Fq)
Quasidihedral_group
Second homology group of a 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
Schur_multiplier
groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is
Z-group
Governing body of association football in Denmark
Landsfodboldturneringen (1913–1927) Provinsmesterskabsturneringen (1913–1931) Sylow-Tournament (1918–1926) The DBU is separated into six regional associations
Danish_Football_Association
proved by John H. Walter (1967, 1969), describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler
Walter_theorem
1990 Japanese film
website Rotten Tomatoes, based on 17 reviews, and an average rating of 7/10. Sylow, Henrik. "Biography". KitanoTakeshi.com. Archived from the original on 16
Boiling_Point_(1990_film)
Concept in mathematics
cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called
Frobenius_group
Country in northern Europe
revolutionary contributions to mathematical logic, while Øystein Ore and Ludwig Sylow advanced group theory. Atle Selberg, a major figure in 20th-century mathematics
Norway
Transport protein
cells". Diabetes. 56 (2): 394–403. doi:10.2337/db06-0823. PMID 17259384. Sylow L, Kleinert M, Pehmøller C, Prats C, Chiu TT, Klip A, et al. (February 2014)
GLUT4
Four finite groups derived from the Leech lattice
properly contains N; hence N is a maximal subgroup of Co0 and contains 2-Sylow subgroups of Co0. N also is the subgroup in Co0 of all matrices with integer
Conway_group
Norwegian military officer and sports official
Carl Christian Weinwich Sylow (26 January 1838 – 12 December 1930) was a Norwegian military officer and sports official. Sylow was born in Christiania
Carl_Sylow
Sporadic simple group
PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J3:2
Janko_group_J3
Sporadic simple group
can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the
Janko_group_J1
Mathematical group that can be generated as the set of powers of a single element
and inductive basis for the representation theory of groups with cyclic Sylow subgroups and more generally the representation theory of blocks of cyclic
Cyclic_group
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Baby_monster_group
Automorphism group of the Klein quartic
class 3A56 generates Sylow 3-subgroup. Any element from the conjugacy classes 7A24, 7B24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a dihedral
PSL(2,7)
Group without normal subgroups other than the trivial group and itself
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
Simple_group
The group P {\displaystyle P} is a Sylow p {\displaystyle p} -subgroup of H {\displaystyle H} , so every Sylow p {\displaystyle p} -subgroup of H {\displaystyle
Frattini's_argument
Transformations induced by a mathematical group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Group_action
Group that is also a differentiable manifold with group operations that are smooth
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Lie_group
Classification theorem in group theory
simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the
Feit–Thompson_theorem
Theorem describing fusion of elements in Sylow subgroup of finite group
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
Focal_subgroup_theorem
Sporadic simple group
O'Nan (1976) in a study of groups with a Sylow 2-subgroup of "Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2nZ ×Z/2nZ
O'Nan_group
theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective
Alperin–Brauer–Gorenstein theorem
Alperin–Brauer–Gorenstein_theorem
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Black_box_group
Number in {..., –2, –1, 0, 1, 2, ...}
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Integer
Type of group
type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups
Locally_finite_group
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Lyons_group
Finite simple group; sometimes classed as sporadic
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Tits_group
Construct in mathematics
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Selmer_group
Abelian group with no non-trivial torsion elements
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Torsion-free_abelian_group
Type of group in mathematics
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Orthogonal_group
of a group Product of group subsets Schur multiplier Semidirect product Sylow theorems Hall subgroup Wreath product Butterfly lemma Center of a group
List_of_group_theory_topics
subgroup H. It states that either G has cyclic or generalized quaternion Sylow 2-subgroups and H contains the centralizer of an involution or G/O(G) has
Strongly_embedded_subgroup
Theorem on the orders of subgroups
hence of a cyclic subgroup, of order any prime dividing the group order. Sylow's theorem extends this to the existence of a subgroup of order equal to the
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Branch of mathematics that studies the properties of groups
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Group_theory
but the case of algebraically closed fields of characteristic p where the Sylow p-subgroups are cyclic is also theoretically approachable. Given a group
Representation_ring
Index of articles associated with the same name
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Janko_group
Group whose operation is composition of permutations
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Permutation_group
Existence of group elements of prime order
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 of p dividing
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
Lie group of complex numbers of unit modulus; topologically a circle
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Circle_group
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Thompson_sporadic_group
Brauer (1964), states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group
Brauer–Suzuki_theorem
Rotation group in 8-dimensional Euclidean space
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
SO(8)
Non-abelian group of order eight
with odd characteristic, the 2-Sylow subgroup of SL2(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized
Quaternion_group
Group of real 2×2 matrices with unit determinant
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
SL2(R)
Number divisible only by 1 and itself
arithmetic progressions as a special case. In the theory of finite groups the Sylow theorems imply that, if a power of a prime number p n {\displaystyle p^{n}}
Prime_number
Operation in group theory
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Semidirect_product
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Fischer_group
Equations of degree 5 or higher cannot be solved by radicals
l'impossibilité de la résolution de l'équation générale du cinquième degré" (PDF), in Sylow, Ludwig; Lie, Sophus (eds.), Œuvres Complètes de Niels Henrik Abel (in French)
Abel–Ruffini_theorem
Monster and modular connection
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Monstrous_moonshine
American mathematician (1927–2021)
Gorenstein, Daniel; Walter, John H. (1962). "On finite groups with dihedral Sylow 2-subgroups". Illinois J. Math. 6 (4): 553–593. doi:10.1215/ijm/1255632706
John_H._Walter
Mathematical finite group theory
whose centralizer has a subnormal subgroup containing t with quaternion Sylow 2-subgroups. Aschbacher, Michael (1977a), "A characterization of Chevalley
Classical_involution_theorem
Orientation-preserving mapping class group of the torus
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Modular_group
Mathematical abelian group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Klein_four-group
simple groups. Finite simple groups of section 2 with rank at least 5 have Sylow 2-subgroups with a self-centralizing normal subgroup of rank at least 3
Gorenstein–Harada_theorem
Decomposition of a number into a product
order to prevent useless ambiguous forms from generating, build up the 2-Sylow group Sll2(Δ) of G(Δ). To obtain an algorithm for factoring any positive
Integer_factorization
Mathematics concept
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Free_group
some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S. J(S) is the Thompson subgroup of a p-group S: the subgroup
ZJ_theorem
generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, (Doerk & Hawkes 1992, I.§6). A subgroup is pronormal if each
Pronormal_subgroup
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Harada–Norton_group
Cardinality of a mathematical group, or of the subgroup generated by an element
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Order_(group_theory)
Mathematical structure with multiplication as its operation
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Multiplicative_group
Theorem in group theory
holds: r = pn p is a Fermat prime and the Sylow 2-subgroups of G are non-abelian and r ≥ pn −pn−1 p = 2 and the Sylow q-subgroups of G are non-abelian for
Hall–Higman_theorem
German mathematician (1849–1917)
Sylow theorems for abstract groups. Earlier proofs had been for permutation groups. His proof of the first Sylow theorem (on the existence of Sylow groups)
Ferdinand_Georg_Frobenius
Set with associative invertible operation
subgroup H {\displaystyle H} divides the order of G {\displaystyle G} . The Sylow theorems give a partial converse. The dihedral group D 4 {\displaystyle
Group_(mathematics)
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Mathieu_group_M22
Type of cyclic group in group theory
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Dicyclic_group
If G is a finitely generated group with exponent n, is G necessarily finite?
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Burnside_problem
Algebraic curve in mathematics
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Elliptic_curve
from G to the abelianization of H. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite
Transfer_(group_theory)
History of a branch of mathematics
since Sylow. This period saw Hans Zassenhaus's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups
History_of_group_theory
Branch of mathematics
emerged, results were reformulated in this abstract setting. For example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite
Abstract_algebra
Sporadic simple group
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Mathieu_group_M12
Set of the values of a function
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Image_(mathematics)
SYLOW
SYLOW
SYLOW
SYLOW
Boy/Male
British, Christian, English
Dark Water; In the Seventeenth Century; Diminutive of Douglas
Boy/Male
Hindu, Indian, Traditional
Shiva
Girl/Female
Muslim
Al-ameeh, Was a great worshipper who worshipped long in the night sometimes right up to dawn (An)
Girl/Female
Italian American
Lady. From the respectful title Donna.
Girl/Female
Arabic, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Sanskrit, Telugu
First Light
Surname or Lastname
English
English : variant of Coombs.
Girl/Female
Arabic, Australian, French
Flower
Girl/Female
Indian, Telugu
Season
Boy/Male
Tamil
Bramhaghosh | பà¯à®°à®®à¯à®¹à®¾à®•à¯‹à®·
Chanting of Vedas
Girl/Female
Australian, Dutch, Finnish, German, Hebrew, Swedish
Gift from God; God is Merciful
SYLOW
SYLOW
SYLOW
SYLOW
SYLOW