Lecture

Symmetric Group and Cycle Decomposition

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In course

MATH-310: Algebra

This is an introduction to modern algebra: groups, rings and fields. ...

Description

This lecture covers the Symmetric group, which is the group of all permutations of a set of elements. It explains the concept of cycle decomposition, where permutations are represented as a product of disjoint cycles. The Orbit-Stabilizer theorem is introduced, showing the relationship between the size of an orbit and the index of the stabilizer subgroup. The lecture also delves into the properties of cycles, transpositions, and the alternating group within the context of permutation groups.

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Instructor

Anna Lachowska

Teaching & PhD Courses Algebra MATH-310 This is an introduction to modern algebra: groups, rings and fields. Analysis I MATH-101(e) We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. Analysis II MATH-106(e) The course studies fundamental concepts of analysis and the calculus of functions of several variables. Representation theory I - finite groups MATH-314 This is a standard c

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Ontological neighbourhood

Mathematics

Algebra: Representation theory

Related concepts (31)

Representation theory of the symmetric group

In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n.

Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .

Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group is a maximal -subgroup of , i.

Symmetric function

In mathematics, a function of variables is symmetric if its value is the same no matter the order of its arguments. For example, a function of two arguments is a symmetric function if and only if for all and such that and are in the domain of The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign under an interchange of variables.

Covering groups of the alternating and symmetric groups

In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in : for n ≥ 4, the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold. For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.

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