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.

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  1. R

    Group Homomorphism & Group Order

    I came across this problem in class note but I was stuck: Assume that ##G## be a group of order 21, assume also that ##G'## is a group of order 35, and let ##\phi## be a homomorphism from ##G## to ##G.'## Assume that ##G## does not have a normal subgroup of order 3. Show that ##\phi (g) = 1##...
  2. G

    Prime factors of binomials

    Homework Statement Is it true that for each ##n\geq 2## there are two primes ##p, q \neq 1## that divide every ##\binom{n}{k}## for ##1\leq k\leq n-1##?Examples: For ##n=6: \binom{6}{1}=6; \binom{6}{2}=15; \binom{6}{3}=20; \binom{6}{4}=15; \binom{6}{5}=6.## So we can have ##p=2## and...
  3. Heisenberg1993

    Basic Questions in linear algebra and group theory

    1- How can infer from the determinant of the matrix if the latter is real or complex? 2- Can we have tensors in an N-dimensional space with indices bigger than N?
  4. maverick280857

    Group Theory query based on Green Schwarz Witten volume 2

    Hi, In chapter 12 of GSW volume 2, the authors remark, "spinors form a representation of SO(n) that does not arise from a representation of GL(2,R)." What do they mean by this? More generally, since SO(n) is a subgroup of GL(2,R) won't every representation of GL(2,R) be a representation of...
  5. F

    Is every diagonalizable representation of a group reducible?

    Hey folks, I'm trying to dip into group theory and got now some questions about irreducibility. A representation D(G) is reducibel iff there is an invariant subspace. Do this imply now that every representation (which is a matrix (GL(N,K)) is reducibel if it is diagonalizable?Best regards
  6. H

    Find All Subgroups of A = {1, 2, 4, 8, 16, 32, 43, 64} | Group Theory Question

    Homework Statement Determine all the subgroups of (A,x_85) justify. where A = {1, 2, 4, 8, 16, 32, 43, 64}.The Attempt at a Solution To determine all of the subgroups of A, we find the distinct subgroups of A. <1> = {1} <2> = {1,2,4..} and so on? <4> = ... ... is this true? are there any other...
  7. Breo

    [Undergraduate/Masters] Group Theory Exercises

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  8. M

    Beginning Group Theory, wondering if subset of nat numbers are groups?

    I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem. I'm learning group theory on my own, and...
  9. Greg Bernhardt

    Group Theory: Definition, Equations, and Examples

    Definition/Summary A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses. The number of elements of a group is...
  10. PsychonautQQ

    The webpage title could be: Subgroups in (R^2,+) with Component-wise Addition

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  11. K

    Is G/H Always an Abelian Group if H is Normal in G?

    Let H be a normal subgroup of G. Then factor group G/H is an abelian subgroup. For x, y not in H xHyH=yHxH and xyH=yxH (xyH)(yxH)^{-1}=id xyx^{-1}y^{-1}=id Are these steps correct? thnx
  12. L

    What Does dad^-1 Equal for Elements in a Left-Coset Outside C_G(a)?

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  13. G

    How to start writing a paper on Number Theory or Group Theory

    Hello :) That's my 2nd year in Math, and I want to start writing an article on NT or Group Theory. I know most of the basic GT and some NT. I still don't know residues/congruences completely, I face problems about understanding the theorems. There are a lot of theorems in these chapters and...
  14. J

    Group theory and quantum mechanics

    How to you get sets of complete basis functions using group theory ? For example , using triangle group for CH3 Cl ?
  15. J

    Free groups: why are they significant in group theory?

    Mathematicians have produced a wide variety of long and complex proofs of the existence of free groups, and there appears to be a strong emphasis upon finding better proofs that involve a variety of techniques. (Examples are http://www.jstor.org/stable/2978086 and "www.jstor.org/stable/2317030"...
  16. U

    Basic Group Theory Proof. Looks easy, might not be.

    Homework Statement Let a,b be elements of a group G. Show that the equation ax=b has unique solution. Homework Equations none really The Attempt at a Solution ax = b . Multiply both sides by a^{-1}. (left multiplication). a is guaranteed to have an inverse since it is an element of a...
  17. Space Pope

    Fields in physics and fields in group theory, are they related?

    I just though of this and though "it's abstract math meeting physics, so probably not". After looking up fields in several abstract algebra books I thought that maybe fields in physics were called as such in physics because they share something with the mathematical structure of fields in group...
  18. R

    Understanding the Symmetry of SU(N) Subgroups in Srednicki's Notation

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  19. L

    Group Theory inner automorphism

    Homework Statement How do I prove that the inner automorphisms is isomorphic to ##S_3##? The attempt at a solution I know ##S_3 = \{f: \{ 1,2,3 \}\to\{ 1,2,3 \}\mid f\text{ is a permutation}\}## and I know for every group there is a map whose center is its kernel so the center of of...
  20. D

    Group Theory Basics for Physics Students

    My prof has been throwing around some group theory terms when talking about spin and isospin (product representations, irreducible representations, SU(3), etc.) I'm looking for a brief intro to group theory, the kind you might find in a first chapter of a physics textbook, so I can get familiar...
  21. T

    Correspondence Theorem in Group Theory

    Hello, I'm following the proof for this theorem in my textbook, and there is one part of it that I can't understand. Hopefully you can help me. Here is the part of the theorem and proof up to where I'm stuck: Let ##N## be a normal subgroup of a group ##G##. Then every subgroup of the...
  22. L

    Group theory question about the N large limit

    Hi! I keep hearing that in the large N limit (so I am talking in specific AdS/CFT but more general too I guess) U(N) and SU(N) are isomorphic. So if I construct, say, the ## \mathcal{N}=1 ## SYM Lagrangian in the large N limit, I can take as gauge group both of the ones mentioned above...
  23. B

    What is the Isomorphism between Groups and its Implications?

    Homework Statement Prove or disprove the following assertion. Let G, H, and K be groups. If G × K \cong H × K, then G \cong H.Homework Equations G × H = \left\{ (g,h): g \in G, h \in H \right\} The Attempt at a Solution I don't even know whether the statement is true or false... I tried...
  24. omephy

    Group Theory Book for QFT - Suggestions?

    I am reading QFT from Srednicki's book. In the 2nd chapter of this book and in the spin half part of this book, group theory and group representation theory is used. Can you suggest me a book from where I can learn this?
  25. M

    Necessity of Group Theory in Particle Physics

    So I'm intending to teach myself some Particle Physics and Standard Model type stuff, I was wondering if someone who's already covered this could give me some advice. I did some Group Theory a few years back and looking over content pages of lecture notes I occasionally spot references to...
  26. E

    [Group Theory] Constructing Cayley Graph from Given Relations

    Homework Statement Show that there exists a group of order 21 having two generators s and t for which s^3 = I and sts^{-1} = t^2. Do this exercise by constructing the graph of the group.Homework Equations Based on the given relations, we have t^7 = I.The Attempt at a Solution Since ##s## and...
  27. TheBigBadBen

    MHB Epimorphisms Between Groups: When is a Homomorphism Onto?

    Interesting question I've happened upon: If there is an epimorphism (i.e. onto homomorphism) $\phi:G\times G \to H\times H$, is there necessarily an epimorphism $\psi:G\to H$? If not, under what conditions can we ascertain such an epimorphism given the existence of $\phi$? I would think that...
  28. N

    Please show me some group theory books

    Please show me some group theory books that considering the combination of quantum mechanics and relativity theory that leads to the needing of notion of fields.I have heard that the irreducible representation of Poicare group leading to the infinite dimensions representation(meaning field...
  29. W

    What's the quickest way to understand group theory in physics?

    I already know about generators, rotations, angular momentum, etc. When I see questions about SO(3), SU(3), and lie groups as it pertains to quantum mechanics, I always hold off on getting into the discussion because I think maybe I don't know what that means. It all seems really familiar...
  30. D

    What are some recommended introductory books on group theory for physicists?

    Hi, I'm interested in doing some self-study this summer and learning some group theory. This has come up a lot as I'm getting into graduate level physics courses, so I'd like a good solid introduction to it. Any recommendations on a book? Preferably one that's at the level of an introductory...
  31. S

    What's the URL for the fantastic group theory wiki?

    I recall visiting a website that was a wiki for group theory and had many articles on specific groups, but I don't find it today doing a simple-simon search on keywords like "group theory". Anyone know the website that I'm talking about?
  32. J

    MHB Group Theory and a Rubik's Cube

    Does anyone know what this guy is on about? I understand some of the basics of group theory and I know there's a connection between Galois theory and the solving of a Rubik's cube, but I'm not sure what law he is even trying to disprove here. I'm assuming something with regards to symmetry or...
  33. M

    MO Diagram from Group Theory: Central Atom

    Homework Statement I am wondering how for determine the central atom's orbitals from the point group character tables described by group theory. For example CO3^-2 (D3h) Carbon's (central atom) p-orbitals are described by a1''+e'. The s-orbital is a1' Homework Equations The...
  34. Mathelogician

    MHB Can Subgroups Form a Group by Union Without Containing Each Other?

    Hi all, Here i ask the fisrt serie of questions i couldn't solve; A basic knowledge of group theory is supposed for solving them! ------------------------------------------------------------ 1- Can you find 3 subgroups H, k and L of a group G such that H U k U L = G ;and no one of the 3...
  35. J

    Simple group theory vocabulary issue

    I am reading about group theory in particle physics and I'm slightly confused about the word "representation". Namely, it is sometimes said that the three lightest quarks form a representation of SU(3), or that the three colors do. But at the same time, it is said that a group can be...
  36. A

    Exploring the Applications of Group Theory in Mathematics and Beyond

    I just studied group theory. Its all nice with all the definitions and rules that are supposed to be followed for a set with a given operation to be called a group. But I fail to see the importance of defining such an algebraic structure. What are its uses?
  37. A

    Proving Subgroup Inclusions in Group Theory | Homework Help

    Homework Statement Here's the question: http://assets.openstudy.com/updates/attachments/511179fae4b0d9aa3c487dfb-ceb105-1360099849081-grouptheory.png Homework Equations The Attempt at a Solution Step 1: <S,T> subset <<S>,T> subset <<S>,<T>> (easy) and Step 2: <S,T> subset...
  38. C

    Proving Subgroups in Finite Groups

    Homework Statement Let G be a finite group, a)Prove that if ##g\,\in\,G,## then ##\langle g \rangle## is a subgroup of ##G##. b)Prove that if ##|G| > 1## is not prime, then ##G## has a subgroup other than itself and the identity. The Attempt at a Solution a) This one I would just like...
  39. D

    Group theory textbook suggestions?

    I'm looking for a text that covers group theory and its applications for QM and QFT, targeted towards an audience that knows their QM but is ignorant of everything quantum fieldy. Any recommendations?
  40. P

    MHB Unique x for all g in G such that $x^m=g$?

    Let G be a group, |G|=n and m an integer such that gcd(m,n)=1. (i) show that $x^m=y^m$ implies $x=y$ (ii)Hence show that for all g in G there is a unique x such that $x^m=g$ (i) there exist a, b such that am+bn=1 so that $m^{-1}=a (mod n)$. Hence $x^m=y^m ->x=y$ ok? (ii) (i) shows...
  41. P

    MHB Exploring Normal Subgroups and p-Groups in Finite Groups

    Let G be a finite group and N a normal subgroup of G. Assume further that N is a p -group for some prime p. 1) By considering G/N, show that there is a subgroup H of G contaning N such that p does not divide [G:H]. 2) Show that N is a subgroup of all p-subgroups of G. My thoughts: for 1)...
  42. S

    Nuclear/Particle Physics & Group Theory: Understanding the Benefits

    I'm pursuing a degree in nuclear physics. However, I have a huge interest in particle physics (i know they are closely related). I am wondering how much a math course in group theory will help me understand particle physics. I want to minor in math, so I'm going to take some extra math...
  43. P

    MHB Proof: G/H1 is Isomorphic to H2/K for G with Normal Subgroups H1 and H2

    Let G be a group with normal subgroups H1 and H2 with H2 not a subset of H1. Let K = H1 intersect H2. Show that if G/H1 is simple, then G/H1 is isomorphic to H2/K. My first thought was to set up a homomorphism with K as the kernel but soon realized that the fact that H2 was not normal is...
  44. H

    Sigma matrices question Group theory

    Homework Statement I have read the following text in a textbook(look the attaxhement) ,and i have a simple question .WHY every 2x2 hermitian matrix would have to satisfy this Equation.It is not obvious to me why.Does anyone know the answer? The textbook stops there without giving any...
  45. G

    Simply group theory problem

    Homework Statement Let (G,.) be an non-abelian group. Choose distinct x and y such that xy≠yx. Show that if x2≠1 then x2\notin{e,x,y,xy,yx} The Attempt at a Solution If x2=x would imply x.x.x-1=x.x-1 and x=e which cannot be. If x2= xy or x2=yx would imply x=y which also cannot be...
  46. G

    How Does xH Equal yH Imply x⁻¹y Belongs to H in Group Theory?

    Homework Statement Let H be a subgroup of G Prove xH=yH ⇔ x-1.y\inH Homework Equations The Attempt at a Solution If x.H = y.H then x,y\inH since H is a subgroup x-1,y-1\inH and the closure of H means x-1.y\inH Proving the reverse is my problem despite the fact that I'm sure...
  47. J

    Finding the star of a wave vector using group theory

    I'm working on a problem where I have to find the little co-group and star of two wave vectors for a diamond structure (space group 227). I know I have to act on the vector by the symmetry operations in the group (perhaps only the ones in the isogonal point group, Oh?) and see if it remains the...
  48. ShayanJ

    Classical physics and Group theory

    You know that the current theories in particle physics are expressed in the language of group theory and the symmetries of the theory describe its properties I don't know how is that but my question is,can we do that to classical physics too? I mean,can we use maxwell's equations and derive a...
  49. A

    Topology, functional analysis, and group theory

    What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.
  50. Chris L T521

    MHB POTW: Proving Group Theory Induction & SO(2, $\mathbb{R}$) Isomorphism

    Thanks to those who participated in last week's POTW! Here's this week's problem (I'm going to give group theory another shot). ----- Problem: (i) Prove, by induction on $k\geq 1$, that \[\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}^k =...
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