What is Group: Definition and 1000 Discussions

In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while being associative as well as having an identity element and inverse elements. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups arise as symmetry groups in geometry but appear also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Point groups describe symmetry in molecular chemistry.
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term of group (groupe, in French) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

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  1. redtree

    I The generators of a ``Poincare-type'' group in momentum space

    Can someone share a paper or chapter from a textbook if they know a good one? I'm curious to see the explicit form of these matrices. In position space, the generators of boosts act on the rapidity, which can be related to velocity in X. Assuming the generators of boosts in K act on rapidity in...
  2. P

    A On the group notation of the 1975 Wu-Yang paper

    In the 1975 Wu–Yang paper on electromagnetism=fiber bundle theory Table 1: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.12.3845 Wu & Yang use the notation ##\mathrm{U}_1(1)## for the bundle of electromagnetism and ##\mathrm{SU}_2## for the isospin gauge field. I am unfamiliar with this...
  3. Structure seeker

    I Multiplication in projective space

    Let #F# be a field and consider the projective space of dimension #n# over it with added the point #0#. It seems to me that there is a valid definition of multiplication by just entrywise multiplicating the elements. Of course both can be multiplied by #x \in F# but that goes for the product as...
  4. A

    A About calculating a fundamental group

    What is the way to compute ##\pi_1(PGL_2(R))##? Is it related to defining an action of ##PGL_2(R)## on ##S^3##? it would be helpful if you can provide me with relevant information regarding this
  5. T

    Chance of Selecting Twins from Group of 30 [Solved!]

    TL;DR Summary: Chance of picking 2 named people when randomly choosing 3 from a group of 30. For my daughter's homework question: There is a group of 12 girls and 18 boys. Two of them are twins (girl and boy). If I select three at random, what is the chance that the twins will be chosen? I...
  6. bella987

    Deriving the commutation relations of the Lie algebra of Lorentz group

    This is the defining generator of the Lorentz group which is then divided into subgroups for rotations and boosts And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps: especially...
  7. L

    A What is the generator of the cyclic group (Z,+)?

    I do not understand why ##(Z.+)## is the cyclic group? What is a generator of ##(Z,+)##? If I take ##<1>## I will get all positive integers. If I take ##<-1>## I will get all negative integers. I should have one element which generates the whole group. What element is this?
  8. 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...
  9. T

    I SO(3) -- What is the advantage of knowing something is in a group?

    Good Morning! I know that Rotation matrices are members of the SO(3) group. I can prove some useful properties about it: The inverse is the transpose; Closure properties; However, what is the advantage of asserting that a rotation matrix is a member of the SO(3) group, when all I really need...
  10. PhysicsRock

    Prove relation between the group of integers and a subgroup

    So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt. $$ 0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
  11. A

    I What group might represent the symmetries of these carbon rings?

    The carbon rings in the upper-middle of this page https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/react3.htm such as corannulene or coronene possess symmetries. But, they are not the typical dihedral arrangements of points, like a single hexagon or single pentagon or single equilateral...
  12. PhysicsRock

    What does the identity mean here?

    I don't understand why the identity is mentioned in the group's definition and how I am supposed to incorporate it into the table. I honestly have missed some lectures on Linear Algebra, and I can't find any examples or definitions for this in the prof's notes. I'd appreciate some help for sure...
  13. P

    I Group like operations that are not associative

    A group can be defined by the following three properties. (Source: wikipedia) Is there any example of an operation that fails the associativity test, but meets the other two tests? I'll refer to this hypothetical entity as an almost-group for the purposes of this post lacking any knowledge...
  14. Ahmed1029

    I What Determines the Speed and Clarity of Signals in Wave Physics?

    What exactly is a signal in wave physics? Is any wave considered a signal? Like, consider a superposition of harmonic plane waves, is the signals it carries considered the envelope(that travels at the group velocity) or the individual rippes that travel at a the phase velocity?
  15. 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...
  16. Z

    I Classifying group homonorphisms

    What is the field of study called that classifies homomophisms of groups?
  17. H

    A Calculate the group velocity in EIT (famous paper: light speed 17m/s)

    hello everyone! Recently,i'm reading a paper about slow light,that's really a famous work published in Nature.[Light speed reduction to 17 metrespersecond in an ultracold atomicgas]. But I'm trouble with some calculation about the velocity of slow light.here are below: i try to use the...
  18. A

    B How does the Milky Way galaxy move in the local Group?

    How does the Milky Way galaxy move in the local Group? Is there a circular motion around the center of the local Group like the sun moves around the center of the galaxy?
  19. Rzbs

    I Space Group Symmetry: What is F d3m Spinel Structure?

    What does mean spinel structure has F d3m space group? I know F is for face centred cubic, 3 is 3-fold symmetry and m is mirror, but I don't know what means "d"?
  20. H

    I Group velocity for an electromagnetic wave inside glass

    Hi, I saw that the group velocity for an electromagnetic wave can be calculate with the following formula ##v_g = v_p + k \frac{d v_p}{dk}## Thus, since ##v_p = \frac{c}{n} = \frac{\omega}{k}## Is it correct to say that ##v_g = \frac{c}{n} + k(- \frac{\omega}{k^2})## where ##k =...
  21. L

    A SO(3) group, Heisenberg Hamiltonian

    We have commutation relation ##[J_j,J_k]=i \epsilon_{jkl}J_l## satisfied for ##2x2##, ##3x3##, ##4x4## matrices. Are in all dimensions these matrices generate ##SO(3)## group? I am confused because I think that maybe for ##4x4## matrices they will generate ##SO(4)## group. For instance for...
  22. 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...
  23. P

    I Is Group Multiplication Possible in Non-Mathematical Topics?

    Sorry if it didnt belong. its not at calculus or linear algebra.
  24. K

    I Generators of Galois group of ## X^n - \theta ##

    As the summary says we have ## f(x) = x^n - \theta \in \mathbb{Q}[x] ##. We will call the pth primitive root ## \omega ## and we denote ##[\mathbb{Q}(\omega) : \mathbb{Q}] = j##. We want to show that the Galois group is generated by ##\sigma, \tau## such that $$ \sigma^j = \tau^p = 1...
  25. redtree

    I Is the spin group spin(n) a double cover for O(n)?

    Every rotation in O(n) seems like it should map to spin(n) even if some rotations are not continuously connected to the identity.
  26. redtree

    I Why are discontinuous Lorentz transformations excluded from the Poincare group?

    The full Lorentz group includes discontinuous transformations, i.e., time inversion and space inversion, which characterize the non-orthochronous and improper Lorentz groups, respectively. However, these groups are excluded from the Poincare group, in which only the proper, orthochronous...
  27. 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...
  28. ergospherical

    Is There a Simple Way to Show GL(n,C) is a Lie Group?

    I'd like to clarify a few things; the approach is basically just to show that ##\mathrm{GL}(n,\mathbf{C})## is isomorphic to a subgroup of ##\mathrm{GL}(2n,\mathbf{R})## which is a smooth manifold (since ##\mathbf{R} \setminus \{0\}## is an open subset of ##\mathbf{R}##, so its pre-image...
  29. P

    Notation clarification: SU(N) group integration

    Hello, I would like help to clarify what det( {\delta \over \delta J}) W(J) (equation 15.79) actually means, and why it returns a number (and not a matrix). This comes from the following problem statement (Kaku, Quantum Field Theory, a Modern Introduction) Naively, one would define det...
  30. Haorong Wu

    I How to find the generator of this Lie group?

    Hello, there. Consider a Lie group operating in a space with points ##X^\iota##. Its elements ##\gamma [ N^i]## are labeled by continuous parameters ##N^i##. Let the action of the group on the space be ##\gamma [N^i] X^\iota=\bar X^\iota (X^\kappa, N^i)##. Then the infinitesimal transformation...
  31. V

    I Does associativity imply bijectivity in group operations?

    Quick question: do the group axioms imply that the group operator is bijective? More in general, does associativity imply bijectivity in general? I can think about a subgroup of S3 that only operates on 2 elements, but it is really isomorphic to S2. But is there some concept or term for a...
  32. joneall

    A Simple definition of Lie group

    I'm writing some notes for myself (to read in my rapidly approaching declining years) and I'm wondering if this statement is correct. I"m not sure I am posting this question in the right place. "Summary: The matrix representations of isometric (distance-preserving) subgroups of the general...
  33. Y

    MHB Group Elements of Z24: Find the Order

    Determine the order of every element of Z24
  34. K

    I Showing that a group acts freely and discretely on real plane

    So before I start I technically do now that the group I am dealing with is just a representation of the Klein bottle but I am not supposed to use that as a fact because the goal of the problem is to derive that information. Problem: Let G be a group of with two generators a and b such that aba...
  35. A

    Mono-esterify a mono-phosphate salt group to a carboxylic acid

    I am interested to see if anyone knows the easiest way to mono-esterify a mono-phosphate salt group to a carboxylic acid. How to facilitate the resultant -> product ? CH3COOH + NaH2PO4 ‐> C2H4NaO5P + H20
  36. fresh_42

    A Center of a linear algebraic group

    Let ##G\leq GL(n)## be a linear algebraic group of dimension ##m,## and ##C## its ##c##-dimensional center. What do we know about lower and upper bounds of ##c=c(m)\,\text{?}## Clearly ##c(0)=0, c(1)=1## and ##n^2\geq c(m)\geq 1## for ##m\neq 0.## By Schur's Lemma we also know ##c(n^2)=1##. Did...
  37. Labyrinth

    A Link between 24 dimension kissing number and Monster group

    I've heard that there is some link between these two values (they're so close!) but I can't seem to find it anywhere. Can someone point me in the right direction? (there's also the J-invariant 196884, well you get the idea)
  38. 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...
  39. A

    I Unitary Representation of Poincaré Group: Classical Relativistic Mechanics

    This thread is a shameless self-promotion of a recent work of mine: https://arxiv.org/abs/2105.13882 In the paper an operational version of classical relativistic dynamics (for massive particles) is obtained from an irreducible representation of the Poincaré group. The formalism has kets...
  40. 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...
  41. W

    I Can a group be isomorphic to one of its quotients?

    Of course it must be an infinite group, otherwise |G/N|=|G|/|N| and then {e} is the only ( and trivial) solution. I understand there is a result that for every quotient Q:=G/N there is a subgroup H that is isomorphic to Q. Is that the case?
  42. Kaguro

    Proving Closure and Identity for U(n)

    Closure Let a,b ∈U(n). a has no common factor with n (other than 1) b has no common factor with n(,,) So, If ab < n, then ab doesn't have any common factors with n. If ab>n, then for some p,ab-pn < n. Since ab doesn't have any common factor with n, ab-pn can't either. (ab≠ n, because neither a...
  43. S

    Group velocity dispersion on two pulses of different lengths

    $$\tau _{01} = 10 \tau _{01}$$ If I calculate ##\frac{\tau_{p1}}{\tau_{p1}}## and set z=d=1cm I do not know how to continue from there as I can't solve the equation without knowledge of τ0 for D. $$\frac{\tau_{p1}}{\tau_{p1}} = \frac{\tau_{02} \cdot 10}{\tau_{02}} \sqrt{\frac{1+\frac{d^2 \cdot 4...
  44. J

    Group Table: Understanding the Identity Element and Avoiding Repeated Values

    * s t u v s s? v? u v? t t v u u? v v? Since s*u=u does that mean s is the identity element? Then I know there can't be repeated values in a row or column so I need to us that to somehow fill in the rest of the blank spaces?
  45. S

    Naming the pentane with phenyl group (or benzene with aryl group)

    I understand the first one, which indicates that there is a phenyl group in the second carbon of pentane. But where did the 2 in 2-pentylbenzene come from? Shouldn't if it be just pentylbenzene, since there is only one subtitute(pentane) on the benzene? If not, would you explain why? Thank you.
  46. F

    I Does mean velocity equal group velocity of wave packets in QM?

    Does mean velocity of particle equal group velocity of wave packet in QM?If they do not equal which of them is classical velocity?
  47. L

    A Understanding the Relationship between Orthogonal and Unitary Groups

    I'm a little bit confused. Matrices \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} ##\theta \in [0,2\pi]## form a group. This is special orthogonal group ##SO(2)##. However it is possible to diagonalize this matrices and get \begin{bmatrix} e^{i\theta} & 0...
  48. mcas

    Check invariance under rotation group in spacetime

    I started by inserting ##ds=\sqrt{dx'^{\mu} dx'_{\mu}}## and ##p'^{\mu}=mc \frac{dx'^{\mu}}{ds}##. So we have: $$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$ Now I know that ##dx'^{\mu}=C_\beta \ ^\mu dx^\beta## and ##dx'_{\mu}=C^\gamma \ _\mu dx_\gamma## where...
  49. L

    I Understanding Unfaithful Representations of Z_2 in the Caley Table

    Every group needs to have that every element appear only once at each row and each column. But in the case of unfaithful representations of ##Z_2## sometimes we have ##D(e)=1##, ##D(g)=1##. When we write the Caley table we will have that one appears twice in both rows and in both columns. How is...