Group Definition and 1000 Threads

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. Mr Davis 97

    I The cases in proving that group of order 90 is not simple

    https://imgur.com/a/FuCPJLe I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that let's us count elements and get a contradiction, but why is the second case there? In other words, why...
  2. Mr Davis 97

    No group of order 10,000 is simple

    Homework Statement Show that there is no simple group of order ##10^4##. Homework EquationsThe Attempt at a Solution By way of contradiction, suppose ##G## is simple and ##|G| = 10000 = 5^42^4##. Sylow theory gives ##|\operatorname{Syl}_2(G)| = 1## or ##16##. If ##|\operatorname{Syl}_2(G)| =...
  3. 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...
  4. A

    Sub groups of the dihedral group

    Homework Statement This is only a step in a proof I am trying to make. Let Dm be the dihedral group. r is the rotation of 2π/m around the origin and s is a reflexion about a line passing trough a vertex and the origin. Let<s> and <r> be two subgroups of Dm. Is there a theorem that states...
  5. J

    Proving Dn with Involutions: Group Representation Homework

    Homework Statement let n ≥ 2 Show that Dn = < a,b | a2, b2, (ab)n> Homework EquationsThe Attempt at a Solution I see that a and b are involutions and therefore are two different reflections of Dn. If we set set b = ar where r is a rotation of 2π/n And Dn = <a,r | a2, rn, (ab)2 > I am unsure...
  6. A

    A Quantum Gravity Research Group -- any standing in mainstream Physics?

    I would like to know if this group http://www.quantumgravityresearch.org/ and its Emergence Theory has any standing in main stream Physics. Thanks Andrew
  7. tomdodd4598

    I Exploring Direct Sums of Lorentz Group Representations

    Hey there, I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...
  8. 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...
  9. Ygggdrasil

    New group of eukaryotes discovered

    https://www.cbc.ca/news/technology/hemimastigotes-supra-kingdom-1.4715823 Citation to the paper being discussed: Lax et al. Hemimastigophora is a novel supra-kingdom-level lineage of eukaryotes. Nature. Published online 14 Nov 2018. https://www.nature.com/articles/s41586-018-0708-8
  10. 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...
  11. Mr Davis 97

    Showing that GL(F_p) is isomorphic to an automorphism group

    Homework Statement Let ##p## be a prime integer. Show that ##\operatorname{Aut}(\underbrace{Z_p\times \dots \times Z_p}_{n \text{ factors }})\cong GL_n(\mathbb{F}_p)##, where ##\mathbb{F}_p## is ##Z_p## viewed as a field. Homework EquationsThe Attempt at a Solution First, note that ##Z_p =...
  12. T

    I Images of elements in a group homomorphism

    Why does the image of elements in a homomorphism depend on the image of 1? Why not the other generators?
  13. K

    I States and Group: Eigenvectors Represent One Dimension

    Suppose a set of basis vectors are eigenvectors of some operator. So they will provide a one dimensional representation of that operator in the vector space?
  14. K

    Vector representation of Lorentz group

    Homework Statement In this problem, we'll construct the ##(\frac{1}{2},\frac{1}{2})## representation which acts on "bi-spinors" ##V_{\alpha\dot{\alpha}}## with ##\alpha=1,2## and ##\dot{\alpha}=1,2##. It is convential, and convenient, to define these bi-spinors so that the first index...
  15. Mr Davis 97

    I Why must the group N be finite in this result?

    Ffom exercise 27 of Dummite and Foote: Let ##N## be a finite subgroup of ##G##. Show that ##gNg^{-1}\subseteq N## if and only if ##gNg^{-1} = N##. Why must the subgroup ##N## be finite? Isn't this result true for subgroups of any size?
  16. Mr Davis 97

    I The quotient group of a group with a presentation

    Suppose that we know that ##G=\langle S \mid R\rangle##, that is, ##G## has a presentation. If ##N\trianglelefteq G##, what can be said about ##G/N##? I know that for example, if ##G=\langle x,y \rangle##, then ##G/N = \langle xN, yN \rangle##. But is there anything that can be said about the...
  17. Mr Davis 97

    Show that given conditions, element is in center of group G

    Homework Statement Let ##G## be a finite group and ##m## a positive integer which is relatively prime to ##|G|##. If ##b\in G## and ##a^mb=ba^m## for all ##a\in G##, show that ##b## is in the center of ##G##. Homework EquationsThe Attempt at a Solution Let ##|G| = n## and ##b\in G##. Note that...
  18. CharlieCW

    General Irreducible Representation of Lorentz Group

    This one may seem a bit long but essentially the problem reduces to some matrix calculations. You may skip the background if you're familiar with Lorentz representations. 1. Homework Statement A Lorentz transformation can be represented by the matrix...
  19. Robin04

    I How to show that commutative matrices form a group?

    Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?
  20. Mr Davis 97

    Proving Cauchy's Theorem in Group Theory

    Homework Statement Let ##S = \{(x_1, \dots, x_p) \mid x_i \in G, x_1 x_2 \cdots x_p = e\}##. Let ##C_p## denote cyclic subgroup of ##S_p## of order ##p## generated by the ##p##-cycle, ##\sigma = (1 \, 2 \, \cdots \, p)##. Show that the following rule gives an action of ##C_p## on ##S## $$...
  21. Cryo

    A Nonlinear susceptibility and group reps

    Dear All short explanation: I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point...
  22. C

    MHB Proving matrix group under addition for associative axiom

    Dear Everyone, I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group: Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same. We want to show that T is an...
  23. Mr Davis 97

    Showing that every finite group has a composition series

    Homework Statement Prove that for any finite group ##G## there exists a sequence of nested subgroups of ##G##, ##\{e\}=N_0\leq N_1\leq \cdots \leq N_n=G## such that for each integer ##i## with ##1\leq i\leq n## we have ##N_{i-1}\trianglelefteq N_i## and the quotient group ##N_i/N_{i-1}## is...
  24. L

    Group Theory: Finite Abelian Groups - An element of order

    Homework Statement Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists. Homework Equations /propositions/definitions[/B] Fundamental Theorem of Finite Abelian Groups Lagrange's Theorem and its corollaries (not sure if helpful for this...
  25. Y

    How to show speed is equal to group velocity?

    Homework Statement My question is, how do I show that speed is equal to group velocity? More information at https://imgur.com/a/m6FwNaG Homework Equations v_g = dw/dk The Attempt at a Solution Part a is substitution, part b uses v_g = dw/dk, part c is multiplication by h-bar, but I am stuck...
  26. Mr Davis 97

    Showing that the alternating group is normal

    Homework Statement For each natural number ##n##, let ##V_n## be the subset of the symmetric group ##S_n## defined by $$V_n = \{(i j)(k l) | i,j,k,l \in \{1,\ldots, n\}, i \neq j, \text{ and } k \neq l\},$$ that is, ##V_n## is the set of all products of two 2-cycles. Let ##A_n## be the...
  27. Mr Davis 97

    Prove that the roots of unity is a cyclic group

    Homework Statement Let ##\mu=\{z\in \mathbb{C} \setminus \{0\} \mid z^n = 1 \text{ for some integer }n \geq 1\}##. Show that ##\mu = \langle z \rangle## for some ##z \in \mu##. Homework EquationsThe Attempt at a Solution My thought would be just to write out all of the elements of ##\mu## in...
  28. karush

    MHB *aa3.2 Let Q be the group of rational numbers under addition

    aa3.2 Let Q be the group of rational numbers under addition and let $Q^∗$ be the group of nonzero rational numbers under multiplication. In $Q$, list the elements in $\langle\frac{1}{2} \rangle$, In ${Q^∗}$ list elements in $\langle\frac{1}{2}\rangle $ ok just had time to post and clueless
  29. Mr Davis 97

    Characterizing subgroups of a cyclic group

    Homework Statement Show that for every subgroup ##H## of cyclic group ##G##, ##H = \langle g^{\frac{|G|}{|H|}}\rangle##. Homework EquationsThe Attempt at a Solution At the moment the most I can see is that ##|H| = |\langle g^{\frac{|G|}{|H|}}\rangle|##. This is because if...
  30. Mr Davis 97

    Showing that Aut(G) is a group

    Homework Statement Prove that, for any group ##G##, the set ##\operatorname{Aut} (G)## is a group under composition of functions. Homework EquationsThe Attempt at a Solution 1) associativity: It is a known fact of set theory that composition of functions is an associative binary operation. 2)...
  31. Mr Davis 97

    Finding the order of element of symmetric group

    Homework Statement Let ##n## be a natural number and let ##\sigma## be an element of the symmetric group ##S_n##. Show that if ##\sigma## is a product of disjoint cycles of orders ##m_1 , \dots , m_k##, then ##|\sigma|## is the least common multiple of ##m_1 , \dots , m_k##. Homework...
  32. Mr Davis 97

    Order of element and order of cyclic group coincide

    Homework Statement Let ##G## be a group and ##x \in G## any element. Prove that if ##|x| = n##, then ##|x| = |\{x^k : k \in \mathbb{Z} \}|##. Homework EquationsThe Attempt at a Solution Let ##H = \{x^k : k \in \mathbb{Z} \}##. I claim that ##H = \{1,x,x^2, \dots , x^{n-1} \}##. First, we show...
  33. Martin T

    Vladimir I. Arnold ODE'S book, about action group

    hi everyone, I'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book...
  34. M

    MHB Which of the group axioms are satisfied?

    Hey! :o I want to check the following sets with the corresponding relations if they satisfy the axioms of groups. $M=\mathbb{R}\cup \{\infty\}$ with the relation $\min:M\times M\rightarrow M$. It holds that $\min (a, \infty)=\min (\infty, a)=a$ for all $a\in M$. $M=n\mathbb{Z}=\{n\cdot...
  35. M

    MHB Show that G is a subset of the symmetric group

    Hey! :o Let $n\in \mathbb{N}$ and $M=\{1, 2, \ldots , n\}\subset \mathbb{N}$. Let $d:M\times M\rightarrow \mathbb{R}$ a map with the property $$\forall x, y\in M : d(x,y)=0\iff x=y$$ Let \begin{equation*}G=\{f: M\rightarrow M \mid \forall x,y\in M : d(x,y)=d\left (f(x), f(y)\right...
  36. C

    I Lorentz Group: Tensor Representation Explained

    I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...
  37. Mr Davis 97

    Infinite cyclic group only has two generators

    Homework Statement Let ##H = \langle x \rangle##. Assume ##|x| = \infty##. Show that if ##H = \langle x^a \rangle## then ##a = \pm 1## Homework EquationsThe Attempt at a Solution Here is my attempt: Suppose that ##H = \langle x^a \rangle##. Then, for arbitrary ##b \in \mathbb{Z}##, ##x^b =...
  38. Mr Davis 97

    I Why Does the Proof Assume ##|x| = n## in Finite Cyclic Groups?

    Problem: If ##H = \langle x \rangle## and ##|H| = n##, then ##x^n=1## and ##1,x,x^2,\dots, x^{n-1}## are all the distinct elements of ##H##. This is just a proposition in my book with a proof following it. What I don't get is the very beginning of the proof: "Let ##|x| = n##. The elements...
  39. Mr Davis 97

    I Using group action to prove a set is a subgroup

    Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##. So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group...
  40. T

    MHB What is the minimal dimension of a complex realising a group representation?

    This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is...
  41. Mr Davis 97

    I Given order for every element in a symmetric group

    Compute the order of each of the elements in the symmetric group ##S_4##. Is the best way to do this just to write out each element's cycle decomposition, or is there a more efficient way?
  42. K

    A Symmetries of a Diamond Unit Cell - Point Group Confusion

    Dear All, I've been recently reading the very clear text of Burns and Glazer entitled Space Groups for Solid State Scientists in the context of my thesis which requires understanding of symmetries of crystals, more specifically symmetries of (approximate triply periodic minimal surfaces)...
  43. N

    Lie Bracket for Group Elements of SU(3)

    Homework Statement Determine the Lie bracket for 2 elements of SU(3). Homework Equations [X,Y] = JXY - JYX where J are the Jacobean matrices The Attempt at a Solution I exponentiated λ1 and λ2 to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector...
  44. M

    MHB Find the probability that each group has an equal amount of odd and even numbers

    A set of numbers 1,2,...,4N gets randomly divided into two groups with equal amount of numbers. Calculate the probability:7 a) Each group has an equal amount of odd and even numbers, b) All numbers that are divisible by N, to fall in only one of the groups, c) All numbers that are divisible by...
  45. Alvan

    I How a group of nucleus can act as a circular aperture

    So I have been learning about measuring nuclei radius using electron diffraction, using sin(theta)=1.22lambda/d, doing some research I found out that is the equation for circular aperture diffraction but I don’t really understand how a group of nucleus can act as a circular aperture. Also is...
  46. S

    I How Do SU(2) and SO(3) Relate to Spinors and Vectors in Physics?

    Hello! I want to make sure I understand the relation between this and rotation (mainly between SU(2) and SO(3), but also in general). Also, I am a physics major, so I apologize if my statements are not very rigorous, but I want to make sure I understand the basic underlying concepts. So SU(2) is...
  47. J

    What's the superposition principle for group action?

    A very simple question: if given a vector ##v(t_0)## and two group functions ##G(t)## and ##G'(t)##, here ##t## is the parameter of time, the two group functions act on ##v(t_0)## simultaneously, then we can get a vector field ##v(t)##, then how to get ##v(t)##?
  48. S

    I What are the best resources for learning about Lorentz group representations?

    Hello! Can someone recommend me some good reading about the Lorentz group and its representations? I want something to go pretty much in all the details (not necessary proofs for all the statements, but most of the properties of the group to be presented). Thank you!
  49. J

    I Other ways to break the Higgs symmetry group

    Our standard model breaks the Higgs Su(2) electroweak symmetry via the Higgs mechanism. In official beyond the standard models. May I know the different lists of models where the Higgs field can be part of larger symmetry group like SU(10) and different ways to break it?
  50. S

    I Understanding 4-Vector Representations in the Lorentz Group

    Hello! I am reading some notes on Lorentz group and at a point it is said that the irreducible representations (IR) of the proper orthochronous Lorentz group are labeled by 2 numbers (as it has rank 2). They describe the 4-vector representation ##D^{(\frac{1}{2},\frac{1}{2})}## and initially I...
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