What is Group theory: Definition and 378 Discussions

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

View More On Wikipedia.org
Recent contents Top users
  1. G

    Group theory and crystal structure

    Hi, I have the following problem : I generate GaAs (zinc blende structure) supercells, and then I want to replace some As atoms by N atoms. Let's say I have fcc conventional cell repeated twice in the x, y and z direction so that I have a total of 64 atoms, 32 of Ga and 32 of As. 8 atoms...
  2. E

    Group Theory Textbooks: Recommended Books & Numerical Examples

    hi, can anybody recommend any good textbooks for group theory? I've just started a 3rd year course called groups and geometry and the lecture notes aren't the greatest of help. The lecturer just writes down some algebra regarding the topics without much notes to support them and doesn't...
  3. dextercioby

    Proofs of Group Theory for Theoretical Physicists - Daniel

    It's always annoying when one finds in books (written by (theoretical) physicists for (theoretical) physics students) statements like those below without a mere cross-reference for a mathematically-rigurous proof. And that's what I'm searching for right now: either point me to some books, or...
  4. D

    Counting Primitive Roots in Finite Fields without Group Theory

    I have the definition that if F is a finite field then a \in F is a primitive root if ord(a) = |F|-1. Now what I don't understand is how exactly are there \phi(|F|-1) primitive roots? (Note: This material is supposed not to use any group theory.)
  5. S

    Understanding the Notation of U(p,q) in Group Theory | Wybourne Book Study

    Hi all. I'm studying Wybourne book on group theory. I didn't understand this expression: U(p,q) I know what U(p+q,C) and U(n) means, but I'm unfamiliar with the notation of the above statement. Thanks in advance. Somy:smile:
  6. H

    How Can Finite Group Theory Problems Be Solved?

    Group Theory, please help! Okay, so I'm stuck on a couple questions from my homework, and any guidance would be much appreciated. 1. Prove that if G is a finite group with an even number of elements, then there is an element x in G such that x is not the identity and x^2 = e. I know...
  7. Norman

    How Many Parameters Define O(N) and SO(N) Groups?

    A little intro: This is for a whirlwind intro to Group Theory as part of another class (QFT) in which we are not proving anything, simply introducing definitions and theorems. We are not using a textbook, simply some notes the professor has written up for this intro and the notes are very...
  8. G

    Help with a group theory proof

    I need some help with what should be a really simple group theory proof, but for some reason I'm hitting my head against a wall on this one. I seem to be missing something simple to get me to the next step. I'm learning this stuff in Swedish, so I'm not sure about all the English words, but...
  9. honestrosewater

    What does Edgar Escultura think of Group Theory?

    From what I've read of his work on FLT, I think Edgar Escultura is brilliant. I wonder why he hasn't gotten the recognition he deserves. Does anyone know where I can find more articles by him?
  10. A

    How can group theory be applied to improve Rubik's cube solutions?

    I've always been fascinated by Rubik's cube. I have developed solutions for it and all the related cubes 2x2, 3x3, 4x4, 5x5. For me the cube it is to group theory (of a partcular type of group) what a slide rule is to real arithmetic. Even "laboratory" might not be too stong a label for...
  11. Oxymoron

    Group Theory Problems: Calculating Order of Quotient Groups

    Question 1 a Find the order of (\mathbb{Z}_4 \times \mathbb{Z}_2) \backslash \langle (2,1) \rangle. Question 1 b Find the order of (\mathbb{Z}_2 \times \mathbb{Z}_4) \backslash \langle (1,1) \rangle.
  12. Oxymoron

    Group Theory: Solving Problems with S_4

    I need some help on solving these sorts of problems in Group theory. Question Consider the group action of S_4 on itself by conjugation. Determine the orbit and the stabilizer of x = (12)(34)
  13. A

    Group Theory Problem: Finite Index of Intersection of Conjugates of H

    Let H be a finite-indexed infinite subgroup of an infinite group G. Suppose: G = \bigcup _{i = 1} ^{k} g_i H then J = \bigcap _{i = 1} ^{k} g_i H g_i ^{-1} is a normal subgroup of G and an intersection of all of the (finitely many) conjugates of H. Show that J has a finite index.
  14. A

    Group Theory Problems: Proving Subgroups and Isomorphisms for Different Groups

    Problem 1 \alpha ,\, \beta \in S_n,\ \alpha \beta = \beta \alpha and \alpha is an n-cycle. Prove that \beta is a power of \alpha. I know that either \beta = \epsilon, or \beta permutes all the elements, but I don't know how to prove that it must specifically be a power of \alpha. Problem...
  15. A

    Understanding the Rotation Group in Quantum Mechanics: A Comprehensive Guide

    I'm trying to get a handle on the rotation group in quantum mechanics. Does anyone have suggestions or links to clear and consise statements of this material. I am looking for a level of about Sakurai. Thanks
  16. M

    Proving G is Abelian: A Group Theory Case

    seeing lots of group theory here after a really long time... let G be a finite group of order n, where n is not divisible by 3. suppose (ab)^3 = a^3 b^3 ,for a, b in G . prove that G is abelian.
  17. A

    Solving the Group Theory Conundrum: Proving 'a' is in Z(G)".

    i've just started out with a course in group theory...here's a question that's been bothering me for a while now... let G be a group and 'a' ,a unique element of order 2 in G. show that a belongs to Z(G). if every element of the group has order 2 this is pretty easy...but that's not the case...
  18. R

    Group Theory Basics: Order & Cyclic Groups

    I lost my notes for the Intro to Group Theory part of my algebra course last year, and need to know a coulple definitions before i go back to uni this year: ORDER of a group, and CYCLIC group. Thanks Ray Veldkamp
  19. G

    Learn Group Theory for Physics: Beginner Guide

    I'd like to learn group theory to understand QM and particle theory, and I looked at several books on discrete mathematics but they didn't mention SU groups. I'm an absolute beginner in group theory or discrete math, but I don't want to spend too much time on materials unrelated to physics. Do...
  20. E

    Why do orthogonal matrices have a specific number of independent parameters?

    What does it mean to say that a n x n orthogonal matrix has n(n-1)/2 independent parameters? And why is this so? Can this be shown using the equation the summation with respect to i of the product aij(aik)= bjk where j,k=1,2,3. And bjk has the property bjk=1 when j=k...
  21. S

    Introduction to Quantum Group Theory for College Math Majors

    I'm looking for a good introduction on quantum group theory for someone who has already had a semester class on group theory. I'll be putting together a short paper (5 pages) on the topic with the intended audience of college math majors. My naive understanding is that symmetry and interaction...
  22. P

    Is there a misprint in this group theory exercise?

    I'm currently making my way through "Groups and Symmetry", by M. A. Armstrong [ISBN 0387966757], and I'm stuck on what seems like it should be a very simple exercise. Page 14, problem 3.8, states: However, I can't make this work unless I change the "or" to an "and". So, is this a misprint...
  23. E

    Group theory: point group of a cube

    what is the point group of a cube?
  24. A

    Group theory in GR and quantum gravity

    In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions: Is the gauge group for gravity defined as the set of all possible Weyl tensors on a general 4D Riemann manifold? Which abstract group maps onto this set? Is it GL(4) or a subgroup of GL(4)? How...
  25. E

    Chemical applications of group theory

    does anybody know of any good websites that explain group theory and symmetry operators in application to chemistry?
  26. W

    Group Theory: Find the order of

    Group Theory: Find the order of ... Hello, This is a question I have been given for homework. I would post this in homework section of the forum, but I need more than help to find a solution - I want to understand it. Explanation and understanding it is much more important to me than whether...
  27. C

    Group Theory Basics: Where Can I Learn More?

    [SOLVED] Group Theory For Dummies I've become interested in learning about Group Theory. I don't know too much but I see it spring up all over the place and would just like to know what it is about and some of the basics. Could some one please point me in the direction of a good resource that...
  28. E

    Could unified theory group AxB unite gravity and standard model in physics?

    Group Theory and Physics... Suppose we have a quantum field theory with a defined Lie Group of n-parameters, then if we calculated the invariants of the Lie Group...could we then determine the Lagrangian of the theory?. That is my opinion i think that given a group for a theory we could know...
Back
Top