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. N

    Regarding a Group Theory Proof on the Order of Elements in a Group

    Homework Statement Prove that for a finite group A, the order of any element in A divides the order of A. Homework Equations The order of an element a of a group A is the smallest positive interger n such that an = 1. The Attempt at a Solution Well, I know that the order of a...
  2. N

    [group theory] Finite presentations and relations <S|R>

    Homework Statement Let G be a finitely presented group. Suppose we have a finite generating set S. Prove that there is a finite set of relations R \subset F_S such that <S|R> is a presentation of G. Homework Equations NA The Attempt at a Solution I don't know how to do this. I think...
  3. V

    Group theory finding order of element and inverse

    Homework Statement Let a and b be elements of a group,with a^2=e , b^6=e and a.b=b^4.a find its order and express its inverse in form of a^m.b^n Homework Equations The Attempt at a Solution (ab)^2=(ab)(ab)=(ab)(b^4.a)=a(b^5)a (ab)^3=a(b^5)a(ab)=a(b^5)(a^2)b=a(b^6)=ae=a it...
  4. L

    General question about orbits (group theory)

    Hi everyone, I was just wondering if you have an element of a G-set which has an orbit of only itself, Say Orbit(e) = {e}, in the set of permutations where G is the same order group of permutations and the operation is conjugation. This is an Orbit of size 1 correct? I was just...
  5. BWV

    Question on tensor notation in group theory

    in the appendix on Group Theory in Zee's book there is a discussion of commutations for SO(3) two questions - does [J^{ij},J^{lk}] = J^{ij}*J^{lk}-J^{lk}*J^{ij}? and there is an expression in the appendix that the commutator equals i(\delta^{ik}J^{jl} ... i don't understand the why...
  6. 1

    Help understanding a group theory proof

    iam currently studying undergraduate abstract algebra and i have reached to the permutation group topic i understand every thing till now but iam having trouble understanding the proof of "IF the identity permutation I of {1,2...n} is represented by m transpositions then m is even" I...
  7. D

    Understanding Group Theory in Physics

    Hi Everyone, I am kind of looking some online text to understand Lie Algebra, Group Theory and so forth. I usually need application (everyday/science context how it is used) and intuition more than mere mathematical definition to understand topics. So I need some text that gives very deep...
  8. A

    Group Theory Books: Learn Group Theory Basics

    Hello, I was reading these notes on supersymmetry, and in the appendix (which as he says is just to establish his conventions), talks about a lot of group theory and stuff that I don't know. Can someone please recommend a book/lecture notes where I can learn this?
  9. AlexChandler

    Question about the use of group theory in QM

    I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said "...
  10. M

    A Question on Semantics Regarding Group Theory

    Homework Statement Is the set of a single element {e} with the multiplication law ee = e a group?Homework Equations none.The Attempt at a Solution Yes, it is a group. But that is not my question. My question is how do you ask the question? If I were face to face with you and wanted to ask you...
  11. B

    Assigning peaks in a Raman spectrum/ group theory

    Homework Statement I need help assigning the peaks in a Raman spectrum of acetylene(ethyne). The peaks are : Wavenumber Contours 3372 OQS 1973 OQS 613 OPRS 2. The attempt at a solution Ethyne has 7 vibrational modes (...
  12. H

    Understanding the Relationship between Group Theory and Physics

    Hello! I´m currently reading 'Groups, Representations and Physics' by H.F. Jones and I have drawn some conclusions that I would like to have confirmed + I have some questions. :) Conclusions: 1. An albelian group has always only one irrep. 2. The direct sum of two representations...
  13. B

    Group theory, order of a product of two elements

    I search for an 'elementary' proof of this, where results about structure of abelian groups are not used. I've tried a standard way of proving this, but hit a wall. I'm mainly interested if my work on a proof can be expanded to a full solution. Homework Statement Let G be an abelian group...
  14. L

    Understanding the Identity Element in Finite Abelian Groups

    This is not homework. Self-study. And I'm really enjoying it. But, as I'm going through this book ("A Book of Abstract Algebra" by Charles C. Pinter) every so often I run into a problem or concept I don't understand. Let G be a finite abelian group, say G = (e,a1, a2, a3,...,an). Prove...
  15. A

    Exploring the Group Theory of Latin Squares

    As I'm studying permutation groups I remembered that when I was in elementary school my teacher introduced Latin squares to us and asked us to find all 4 by 4 Latin squares. I never succeeded in solving the problem and I found it so challenging at that time, even later in high school when I...
  16. B

    Group Theory: Ord(x) = Ord(g^-1xg)

    Homework Statement If x and g are elements of the group G, prove that |x|=|g-1xg|. Deduce that |ab| =|ba| for all a,b \in GHomework Equations The Attempt at a Solution I've written a proof but does not seem quite right: if (g-1xg)n= xm then n must be equal m g-nxngn = xm then xngn =gnxm xn...
  17. C

    Conventional Operators in Group Theory Homework

    Homework Statement I've just started to study group theory, and i keep encountering questions where no operators are specified so i was wondering if there was a conventional operator that was meant to be used. For instance I had a question to prove that a cyclic group of order 14 is isomorphic...
  18. L

    Anyone know of a good website that reviews group theory?

    I have never taken an abstract algebra course. So now that in inorganic chemistry they are throwing group theory at us, I am really confused. It all seems like a jumbled mess of random relationships and numbers. I am fine with point groups. Symmetry comes easily to me, so that is not an...
  19. A

    A syllabus of books/literature from calculus to group theory to QFTs

    The study of particles and fields is not made easier by all of the seemingly disparate physics ideas and mathematical methods. I put together an 8 page syllabus of math and physics books, as well as reference literature, to take you from a junior level math or physics background to...
  20. E

    Proving (ab)^n=e: A Group Theory Question | Homework Statement

    Homework Statement If a and b are in a group, show that if (ab)^n=e then (ba)^n=e. Homework Equations The Attempt at a Solution I'm not sure how one would prove this. The question is obviously for non-abelian groups.
  21. E

    Permutations? and Group Theory

    Homework Statement In a finite group, show that the number of nonidentity elements that satisfy the equation x^5=e is a multiple of 4. If the stipulation that the group is finite is omitted, what can you say about the number of nonidentity elements that satisfy the equation x^5=e? Homework...
  22. S

    Quick group theory proof with some minor inquiries

    Homework Statement See attachment, problem #19a. Homework Equations The Attempt at a Solution a) Let j ∈ S X S be arbitrary. Then j is an ordered pair of the form (a,b) for some a,b ∈ S. Now let c = a + b + ab ∈ S. Then clearly a*b = c. Now let d ∈ S and assume a*b = d. But...
  23. B

    Elementary Group Theory, p-group automorphism.

    I am self-studying elementary abstract algebra over the summer, with the book "Abstract Algebra: An Introduction" by Hungerford. It's my first exposure to mathematical proofs (well, short of an introduction in my intro to discrete mathematics), so sometimes I'm not really sure I do everything...
  24. P

    Group Theory (Abstract algebra manipulations)

    Homework Statement Let a,b,c,d be elements of a group G and let ab = c, bc = d, cd = a, da = b. Examine the expression da^2b and first derive an expression for b in powers of a. Then express c and d in powers of a. Show that a^5 = e (identity element) Homework Equations The Attempt...
  25. W

    Cornwell's Group Theory: Read Reviews & Buy

    Has anyone read the book, https://www.amazon.com/dp/0121898008/?tag=pfamazon01-20? It looks good on rough browsing.
  26. N

    What is the objective of renormalization group theory?

    Please teach me this: It seem to me that the objective of renormalization were the exclusion the infinities.But in renormalization group theory,they consider the dependence of physics parameters(e.g the interaction constant lamda,the mass parameter) on momentum p.Then I do not understand what...
  27. B

    What are the applications of group theory?

    Outside of pure mathematics, where is group theory used?
  28. C

    Books to help with math in Jackson and group theory

    So I'm currently self-studying Jackson's Electrodynamics. The math for my undergrad physics was no problem at all for me as I had a strong background in DEs, PDEs, linear algebra, etc. I haven't looked too much into Jackson so far, but it seems I'm having the most difficulty is just keeping up...
  29. J

    Group theory - triangle rotation matrix problem? probably simple?

    I'm attempting to do some problems in a group theory exercise for the first time and am falling flat on my face. Here's the problem: "the molecule 'triangulum' consists of 3 identical atoms arranged in an equilateral triangle. Using a basis which consists of a single localised orbital on each...
  30. U

    Group Theory: Rotational Symmetries

    Homework Statement Show that the group R of rotational symmetries of a dodecahedron is simple and has order 60. The Attempt at a Solution I see how to get order 60 using the orbit stabilizer theorem. Letting R act in the natural way on the set of faces, we find the size of the orbit...
  31. E

    Math books: group theory and topology

    Just want to ask for recommendations for good math books on 1) groups, modules, rings - all the basic algebra stuff but for a physicist 2) topological spaces, compactness, ... I need books for a theoretical physicist to read up on these topics so that I could study, say, algebraic...
  32. M

    Proving Normality in Group Theory: Guide and Tips for Proving H is Normal in G

    Please can someone help me to prove this If N is a cyclic normal subgroup of G and H is a subgroup of N then H is normal in G.
  33. G

    Representations in group theory

    The group theory course I'm taking is driving me crazy. It's a mandatory class in my undergraduate physics studies but it's all very alien and very abstract to me and my books scarcely give any examples when introducing new concepts. It's just so much harder than calculus or physics courses...
  34. A

    Group Theory, unitary representation and positive eigenvalues

    Hi, I'm new in this forum. I have a problem i can't solve and searching on Google i couldn't find anything. It says: If D(g) is a representation of a finite group of order n , show that K = \sum^{i=1}_{n} D^{\dagger} (g_i) D(g_i) has the properties: b) All eigenvalues of K are...
  35. P

    Group Theory Problem: Proving o(H) is Multiple of n

    Homework Statement this is taken from herstein topics in algebra book. G is a finite group such that n divides o(G) define the set H={x/ x^n=e} (is not always a group), prove that the number of elements in H is a multiple of n. o(G) is the number of elements in G Homework Equations uhmm...
  36. C

    The Commutator in Group Theory vs QM

    The commutator for group theory is [X,Y]=X^{-1}Y^{-1}XY whereas the quantum commutator is [X,Y]=XY-YX . At first glance, the two commutators seem to be totally unrelated because the quantum commutator speaks of two binary operations whereas group theory has one binary operation. However...
  37. Z

    Prerequisites for Group Theory

    What concepts should I be familiar with in order to get anything meaningful out of a textbook on group theory? I've read a few articles that talk about a few of group theory's aims and subjects, and it's enough to pique my interest. I've taken first-year Calculus and a bit of Analysis (the...
  38. H

    Where can I find a book on group theory in physics?

    please prefer me a book or booklet or ...that at least includes one section or more about the subject of group theory in physics as i can understand the elements considered in this book about group theory: "Mathematics method for physisists by George Arfken" I'd like it was elementary and...
  39. R

    Why Group Theory Works: SU(6) & Quarks

    Why is it that the group SU(6) gives you the correct wave function for hadrons made up of the u, d, and s quarks? The 6 elements in the fundamental representation would be u up, u down; d up, d down; s up, s down. What's the meaning behind SU(6)? Also, for 6 quarks, would it be SU(12)?
  40. T

    A problem in Finite Group Theory

    This is a problem I encountered in Martin Isaacs' 'Finite Group Theory'. It's located at the end of Chapter II which deals with subnormality, and the particular paragraph is concerned with a couple of not so well-known results which I quote for reference: (In what follows F is the Fitting...
  41. H

    Group Theory: Proving Subgroup of Elements of Order 2 and e

    Homework Statement G is a commutative group, prove that the elements of order 2 and the identity element e form a subgroup. Homework Equations The Attempt at a Solution I don't know where to even begin.
  42. T

    Looking for books on group theory and differential geometry

    My university doesn't offer many courses on theoretical physics (I'm studying applied physics), but because I might want to get my masters degree in theoretical physics, I want to read into some of the math and physics. What books would you recommend to a student who has had linear algebra...
  43. D

    Group Theory: Prove o(b) even if ab = b^-1a

    Group Theory: Prove o(b)|2 if ab = b^-1a Homework Statement Suppose G is a group and a, b \in G a) If o(a) is odd and a*b = b^−1*a, prove that o(b)|2. b) If o(a) is even and a*b = b^−1*a, does it follow that o(b)|2? Prove your answer. Homework Equations n/a The Attempt at a...
  44. P

    Group Theory and Energy Eigenstates in Quantum Mechanics

    I have only recently begun to study group theory from Lie Algebra for Particle Physicists by Georgi. I am slightly confused about the language used by physicists. What does it mean when the following is stated "The energy eigenstates transform like irreducible representations of the group...
  45. V

    How Are Multiplets in Group Theory Related to Spin States in Quantum Mechanics?

    hello, does anyone know what relationship exists, if exists, between the concept of multiplets in group theory and the multiplet as spin state of a system of particles? Thanks, valleyman
  46. S

    Where Can I Find a Good Reference Book on Group Theory in Physics?

    Hello, I'm reposting this in the current section as I'm looking not only for help with homework assignment, but because I'm also looking for good reference textbook. I'm taking a course on group theory in physics, but the teacher is really bad at making the bridge between the maths and the...
  47. micromass

    Group Theory: Is A a Left Coset of G?

    This is not a homework problem. I was just wondering. Let G be a group and let A be a finite subset of G. If |A²|=|A|² (where A^2=\{a_1a_2~\vert~a_1,a_2\in A\} ). Is it true that A is a left coset of G? If A has two elements, then I have proven that this is true. But for greater elements...
  48. E

    Elementary cyclic normal group theory

    Homework Statement If G is a finite group and let H be a normal subgroup of G with finite index m=[G:H]. Show that a^m\in H for all a\in G. Homework Equations order of a group equal the order of element. The Attempt at a Solution no idea.
  49. B

    Short problem on group theory q.3

    urgent due in 13 hrs]short problem on group theory q.3 [due now
  50. B

    Short problem on group theory q.2

    short problem on group theory q.2[urgent due in 13 hrs] due now
Back
Top