What is Group theory: Definition and 375 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.

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  1. I

    Help with a simple group theory question please

  2. I

    What is meant by compex dimension? (Abstract algebra)

    picture since the text is a little hard to read i have no problem showing this is a vector space, but what is meant by complex dimention? Is it just the number on independant complex numbers, so n?
  3. PragmaticYak

    Fixed point free automorphism of order 2

    I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
  4. caffeinemachine

    I Full Course in Group Theory (and More) on YouTube

    I created a YouTube channel (here's the link) a few months ago in which I post detailed lectures in higher mathematics. I just finished my Group Theory Course. Here is a sample video. Apart from that, so far I have uploaded A first course on Linear Algebra (which I am currently renovating). A...
  5. redtree

    I Pin & Spin Groups: Double Covers of Orthogonal & SO Groups

    Pin Groups are the double cover of the Orthogonal Group and Spin Groups are the double cover of the Special Orthogonal Group. Both sets of the double cover are considered to be groups, but it seems that only one of the sets of the double cover actually contains the identity element, which means...
  6. malawi_glenn

    Ideas for group theory for high school math project

    Hi As high school teacher, I sometimes have those extremely talanted and self driven pupils. In their final year, they are required to make a science or math project, roughly one month full-time studies, approx 15-20 pages report. This academic year, one of my students have learned some group...
  7. Group Theory — Introduction to Higher Mathematics

    Group Theory — Introduction to Higher Mathematics

    If you have always wondered what group theory is useful for and why it even exists, this is the video for you. We cover everything from the basic history of group theory, over how and why subgroups partition groups, to the classification of all groups of prime order.
  8. redtree

    B One-to-many relations in group theory

    I apologize for the simple question, but it has been bothering me. One can write a relationship between groups, such as for example between Spin##(n)## and SO##(n)## as follows: \begin{equation} 1 \rightarrow \{-1,+1 \} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n) \rightarrow 1...
  9. M

    Studying Should I study Topology or Group Theory?

    Hello! I'm a physics graduate who is interested to work in Mathematical Physics. I haven't taken any specialized maths courses in undergrad, and currently I have some time to self-learn. I have finished studying Real Analysis from "Understanding Analysis - Stephen Abbott" and I'm currently...
  10. samantha_allen

    Courses Should I take a group theory course before QFT?

    I know that studying QFT requires understanding Lie Groups and infinitesimal generators as they correspond to symmetry transformations. I want to study or take a course (offered by my university) in QFT in the coming academic year and I have the option to take a abstract algebra course offered...
  11. X

    Proper Lorentz transformations from group theory?

    Hi, I was looking at this derivation https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates and I was wondering 1- where does the group structure come from? The principle of relativity? or viceversa? or what? 2- why only linear transformations? I remember...
  12. C

    Direct product of two semi-direct products

    After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$ Any help appreciated.
  13. StenEdeback

    Good introductory book about Lie Group Theory?

    Summary:: Good introductory book about Group Theory? Hi, I am looking for a good introductory book about Group Theory for physicists.
  14. S

    I Degrees of Freedom of SO(3)

    The group ##\rm{O(3)}## is the group of orthogonal ##3 \times 3## matrices with nine elements and dimension three which is constrained by the condition, $$a_{ik}a_{kj} = \delta_{ij}$$ where ##a_{ik}## are elements of the matrix ##\rm{A} \in O(3)##. This condition gives six constraints (can be...
  15. T

    I On group theory

    Hello there.Questions I have: what is the value of group theory?I am not trying to say that it is not important I want to know what made mathematicians study these objects and we still study them today.I know there are very interesting for me at least examples of groups like the Lie group but...
  16. joneall

    A Symmetry of QED interaction Lagrangian

    I am trying to get a foothold on QFT using several books (Lancaster & Blundell, Klauber, Schwichtenberg, Jeevanjee), but sometimes have trouble seeing the forest for all the trees. My problem concerns the equation of QED in the form $$ \mathcal{L}_{Dirac+Proca+int} = \bar{\Psi} ( i \gamma_{\mu}...
  17. patric44

    Solving a Group theory problem using Cayley diagrams

    hi guys i saw this problem : if G is a group and a,b belongs to G and O(a) = e , b.a =a.b^2 then find O(b) , but i want to tackle this problem using Cayley diagrams , so my attempt is as following : $$ba =ab^{2}$$ then i might assume b as flipping , a as rotation : $$ fr = rf^{2}$$ then...
  18. penroseandpaper

    Group theory with addition, multiplication and division

    Hi everyone, I'm working through some group theory questions online. But unfortunately they don't have answers to go with them. So, I'm hoping you can say if I'm on the right track. If this is a binary operation on ℝ, am I right in thinking it satisfies the closure and associativity axioms...
  19. Svend

    Algebra Find the Perfect Group Theory Book for Physicists

    I have failed a course on group theory for physicists in my university, and i need a good book to learn group theory from because anthony zee's book is simply too hard to read. His book is verbose, glosses over many concepts, and is not very rigorous. Then the exercises in the book are very...
  20. P

    MHB Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order

    In the context of group theory, there's a theorem that states that for a given positive integer \(n\) there exist finitely different types of groups of order \(n\). Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this...
  21. E

    Group Theory Sos Due Monday

  22. Haorong Wu

    I Need help with tensors and group theory

    I am reading Group Theory in a Nutshell for Physicists by A. Zee. I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N). It reads I can understand why ##D\left ( R \right )## is a representation of SO(3), but I hardly can see why the tensor T...
  23. sophiatev

    I The SO(3) group in Group Theory

    In Griffith's Introduction to Elementary Particles, he provides a very cursory introduction to group theory at the start of chapter four, which discusses symmetries. He introduces SO(n) as "the group of real, orthogonal, n x n matrices of determinant 1 is SO(n); SO(n) may be thought of as the...
  24. J

    I Group Theory Appearing in Griffith's Elementary Particles (2nd Ed.)

    Hello, I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles. I have had a fair amount of exposure to elementary group theory, but no representation theory, and have some specific questions related to this which refer to the...
  25. V

    I Group Theory sub algebra of unitary group of U(6) group.

    three sub algebra of Unitary group (6) as 1. U(5) . 2. SU(3) 3. O(6) here the three chains in attachment is attached. I want to know how these chains are understands in group theory.
  26. RicardoMP

    A Decomposing SU(4) into SU(3) x U(1)

    I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states. How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...
  27. Adesh

    Understanding the notation in Group Theory

    I was studying mathematical logic and came across this statement of group theory I'm having a hard time in understanding it. I have concluded that ##G## is any set but not an empty one, ##\circ## is a function having input as two variables (both variables are from set...
  28. lelouch_v1

    I Quark Model Families and Masses

    Consider the pseudoscalar and vector meson family, as well as the baryon J = 1/2 family and baryon J = 3/2 family. Within each multiplet, for each particle state write down its complete set of quantum numbers, its mass, and its quark state content. Furthermore, for each multiplet draw the (Y...
  29. Jelly-bean

    I Why do we need two representations of SU(3)

    Summary: if we use up, down and staring quarks and their own antiparticle we can create the Eightfold way and understand mesons by the hyper charge and isospin projections. I don't understand how the conjugate representation of SU(3) allows us to create a vector space of dimension 3, while...
  30. D

    I Quantum number and group theory

    Hi all, Group theory show us that irreducible representation of SO(3) have dimension 2j+1. So we expect to see state with 2j+1 degeneracy. But does group theory help to understand the principle quantum number n ? And in the case of problems with SO(3) symmetry does it explain its strange link...
  31. J

    Algebra Book on how to write proper proofs in Group Theory

    I am trying to learn group theory on my own from Schaum's Outline of Group Theory. I chose this book because there are a lot of exercises with solutions, but I have several problems with it. 1) In many cases the author just makes some handwavey statement and I have to spend hours or days trying...
  32. J

    I Trying to get the point of some Group Theory Lemmas

    There are two related Lemmas in Schaum's Outline of Group Theory, Chapter 4 that seem excessively convoluted. Either I am missing something or they can be made much simpler and clearer. Lemma 4.2: If H is a subgroup of G and {\rm{X}} \subseteq {\rm{H}} then {\rm{H}} \supseteq \left\{...
  33. D

    Other Textbooks for tensors and group theory

    Hello, I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course. I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...
  34. J

    I Mistake in Schaum's Group Theory?

    Schaum's Outline of Group Theory, Section 3.6e defines {{\rm{L}}_n}\left( {V,F} \right) as the set of all one to one linear transformations of V, the vector space of dimension n over field F. It then says "{{\rm{L}}_n}\left( {V,F} \right) \subseteq {S_V}, clearly". ({S_V} here means the set...
  35. S

    A Selection rules using Group Theory: many body

    Hello, I am newish in group theory so sorry if anything in the following is not entirely correct. In general, one can anticipate if a matrix element <i|O|j> is zero or not by seeing if O|j> shares any irreducible representation with |i>. I know how to reduce to IRs the former product but I...
  36. A

    A Create Hamiltonians in condensed matter with group theory

    Hello, I am currently struggling to understand how one can write a Hamiltonian using group theory and change its form according to the symmetry of the system that is considered. The main issue is of course that I have no real experience in using group theory. So to make my question a bit less...
  37. pallab

    Good books on Group theory in High Energy physics

    please suggest me a good book on the high energy physics where group theory is discussed for the beginner.
  38. D

    I Relation Between Cross Product and Infinitesimal Rotations

    Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example: $$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...
  39. diegzumillo

    A Geometric Group Theory

    Hey all I previously asked about some math structure fulfilling some requirements and didn't get much out of it ( Graph or lattice topology discretization ). It was a vague question, granted. Anyway, I seem to have stumbled upon something interesting called geometric group theory. It looks...
  40. N

    Clebsch-Gordan Decomposition for 6 x 3

    Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...
  41. W

    Other Using Michael Artin's "Algebra" for Group Theory

    Hi all, I have stumbled upon Artin's book "Algebra" and was wondering if I could use it to do some self-study on Group Theory. Some background: I am a physics undergraduate who has some competence in elementary logic, proofs and linear algebra. It seemed to me that ideas related to Group...
  42. J

    A Fields transforming in the adjoint representation?

    Hi! I'm doing my master thesis in AdS/CFT and I've read several times that "Fields transforms in the adjoint representation" or "Fields transforms in the fundamental representation". I've had courses in Advanced mathematics (where I studied Group theory) and QFTs, but I don't understand (or...
  43. J

    Show that ##G\simeq \mathbb{Z}/2p\mathbb{Z}##

    Homework Statement Let ##G## be a group of order ##2p## with p a prime and odd number. a) We suppose ##G## as abelian. Show that ##G \simeq \mathbb{Z}/2p\mathbb{Z}## Homework Equations The Attempt at a Solution Intuitively I see why but I would like some suggestion of what trajectory I could...
  44. A

    Commutator group in the center of a group

    Homework Statement [G,G] is the commutator group. Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##. Homework Equations The Attempt at a Solution In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that...
  45. Alex Langevub

    An exercise with the third isomorphism theorem in group theory

    Homework Statement Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##. a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H## b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K## Homework Equations The three...
  46. Auto-Didact

    A Taxonomy of Theories in Theoretical Physics

    It goes without saying that theoretical physics has over the years become overrun with countless distinct - yet sometimes curiously very similar - theories, in some cases even dozens of directly competing theories. Within the foundations things can get far worse once we start to run into...
  47. A

    Isomorphism of dihedral with a semi-direct product

    Homework Statement Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2 $$ where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$ Homework Equations I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this...
  48. Gerson J Ferreira

    Solid State Group theory paper suggestions for my classes

    I teach group theory for physicists, and I like to teach it following some papers. In general my students work with condensed matter, so I discuss group theory following these papers: [1] Group Theory and Normal Modes, American Journal of Physics 36, 529 (1968) [2] Nonsymmorphic Symmetries and...
  49. A

    Show injectivity, surjectivity and kernel of groups

    Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...
  50. T

    I How to properly understand finite group theory

    I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with...