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In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
In mathematics, the free groupFS 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. The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses.
In mathematics, class field theory is the branch of algebraic number theory concerned with describing the Galois extensions of local and global fields. Hilbert is often credited for the notion of class field. But it was already familiar for Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. This theory has its origins in the proof of quadratic reciprocity by Gauss at the end of the 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory.
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. In symbols, a perfect group is one such that G(1) = G, or equivalently one such that Gab = {1}.
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements.
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.
In mathematics, a quasisimple 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
In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
Michio Suzuki was a Japanese mathematician who studied group theory.
The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure.
In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem and the classification of finite simple groups. Several important infinite groups are CA-groups, such as free groups, Tarski monsters, and some Burnside groups, and the locally finite CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-group.
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:
A 2-transitive group is a transitive group used in group theory in which the stabilizer subgroup of every point acts transitively on the remaining points. Equivalently, a group acts 2-transitively on a set if it acts transitively on the set of distinct ordered pairs . That is, assuming that acts on the left of , for each pair of pairs with and , there exists a such that . Equivalently, and , since the induced action on the distinct set of pairs is .
In abstract 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 and is the "first major application of the transfer" according to. The focal subgroup theorem relates the ideas of transfer and fusion such as described in. Various applications of these ideas include local criteria for p-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index p.
In group theory, Bender's method is a method introduced by Bender (1970) for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups Bender (1970b), and Gorenstein and Walter's classification of groups with dihedral Sylow 2-subgroups. Bender's method involves studying a maximal subgroup M containing the centralizer of an involution, and its generalized Fitting subgroup F*(M).
In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson, gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Brauer & Fowler (1955).
In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order
References
- Berkovich, Yakov (2008), Groups of prime power order. Vol. 1, de Gruyter Expositions in Mathematics, 46, Walter de Gruyter GmbH & Co. KG, Berlin, doi:10.1515/9783110208221.285, ISBN 978-3-11-020418-6, MR 2464640
- Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC 154682311
- Conway, John Horton; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
- Hall, Philip (1940), "The classification of prime-power groups", Journal für die reine und angewandte Mathematik , 1940 (182): 130–141, doi:10.1515/crll.1940.182.130, ISSN 0075-4102, MR 0003389, S2CID 122817195
- Struik, Ruth Rebekka (1960). "A note on prime-power groups". Canadian Mathematical Bulletin . 3: 27–30. doi: 10.4153/cmb-1960-006-5 . ISSN 0008-4395. MR 0148744.
- Suzuki, Michio (1982), Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 247, Berlin, New York: Springer-Verlag, ISBN 978-3-540-10915-0, MR 0648772
- Suzuki, Michio (1986), Group theory. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 248, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-86885-6, ISBN 978-0-387-10916-9, MR 0815926