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. RJLiberator

    Group Theory: g^n = e proof

    Homework Statement If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element. Homework EquationsThe Attempt at a Solution I feel like there is missing information, but that cannot be. This seems too simple: The order of G is the smallest possible integer n...
  2. M

    I Sylow subgroup of some factor group

    Hi. I have the following question: Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N? Here is what I think. Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N. Now...
  3. A

    I What are the irreducible representations of point groups and how do I find them?

    As part of physical chemistry I am reading up group theory for molecular symmetries. I realize the way chemistry textbooks treat this must be very different from what mathematicians do. So I want to know how I take a point group, find the matrix operations and get the character table.For an C2...
  4. polyChron

    Rotations in Bloch Sphere about an arbitrary axis

    Hey, (I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!) I am trying to understand the following statement. "Suppose a single qubit has a state represented by the...
  5. P

    Proof involving group of permutations of {1,2,3,4}.

    Homework Statement Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##: ##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...
  6. T

    Inverse image of a homomorphism

    The question: Let f: G -> H be a homomorphism of groups with ker(f) finite, the number of elements being n. Show that the inverse image is either empty or has exactly n elements. My work so far: Let h be eH (identity on H). Then the inverse image is ker(f) so has n elements, which makes it...
  7. applestrudle

    Group Theory why transformations of Hamiltonian are unitary?

    This is what I have so far: I'm trying to show that the matrix D has to be unitary. It is the matrix that transforms the wavefunction.
  8. T

    Deciding between group theory vs. complex analysis

    Hi, I have one spot remaining to take a pure math course, and I'm trying to decide between complex analysis and group theory. Although I've touched some of the basic of dealing with complex numbers in my physics/DE courses, they haven't gone in much depth into them beyond applications. On the...
  9. G

    Cyclic Quotient Group: Is My Reasoning Sound?

    Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...
  10. dsatkas

    Algebra Math textbooks for physics grad student and other questions

    I hope this post won't become too tedious. I've completed my undergrad studies in physics and if things go well i will begin my master's degree in April. The thing is, since my path to graduation has been peculiar (to say the least) I'm kinda weak in maths skills atm and need to improve. I'm...
  11. S

    High Energy Textbook for relativistic quantum mechanics and group theory

    Hi, I am looking for textbooks in relativistic quantum mechanics and group theory. I have just finished my undergraduate studies in Physics and am looking to specialise in theoretical high-energy physics. Therefore, textbooks in relativistic quantum mechanics and group theory suited for that...
  12. DeldotB

    Show a group is a semi direct product

    Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...
  13. sa1988

    Group Theory: Is the following a valid Group?

    Homework Statement Is the following a valid group? The values contained in the set of all real numbers ℝ, under an operation ◊ such that x◊y = x+y-1 Homework Equations Axioms of group theory: Closure Associativity There must exist one identity element 'e' such that ex=x for all x There...
  14. DeldotB

    Using the Second Isomorphism (Diamond Isomorphism) Theorem

    Homework Statement Good day all, Im completely stumped on how to show this: |AN|=(|A||N|/A intersect N|) Here: A and N are subgroups in G and N is a normal subgroup. I denote the order on N by |N| Homework Equations [/B] Second Isomorphism TheoremThe Attempt at a Solution Well, I know...
  15. sa1988

    My first exercise on Group Theory

    EDIT: I've just realized this is the 'Calculus and beyond' subforum - I saw 'beyond' and thought, "Well I've done all my calculus, and now I'm doing group theory, so this thread must go here!". But now I realize it surely belongs somewhere else. Sorry about that. Mods feel free to shift it to...
  16. davidbenari

    Algebra Group Theory for Physics: Weyl vs Herstein?

    I study physics and currently taking a mathematical physics course. One of the topics is group theory and we will see the following topics: Symmetries, discrete groups, homomorphisms, isomorphisms, continuous groups, and linear transformations in phase space. This topic will be covered with...
  17. davidbenari

    Relation of Noether's theorem and group theory

    I'm doing a small research project on group theory and its applications. The topic I wanted to investigate was Noether's theorem. I've only seen the easy proofs regarding translational symmetry, time symmetry and rotational symmetry (I'll post a link to illustrate what I mean by "the easy...
  18. alexmahone

    MHB Proving $(G,*)$ is a Group: Hints and Tips for a Simple Group Theory Problem

    Let $G$ be a set and $*$ a binary operation on $G$ that satisfies the following properties: (a) $*$ is associative, (b) There is an element $e\in G$ such that $e*a=a$ for all $a\in G$, (c) For every $a\in G$, there is some $b\in G$ such that $b*a=e$. Prove that $(G, *)$ is a group. My...
  19. L

    MHB Group Theory: Proving Finite Rank of Torsion-free Abelian Groups Using Independence

    If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...
  20. Andre' Quanta

    Two theorems of Group Theory

    I am studying Group Theory at the moment and i am not sure about a theorem. Is it true that a Lie Group G is compact if and only if every finite complex representation of it is unitary? I know that is true the if, but what about the viceversa? Same question. Is it true that a Lie group is...
  21. B

    Want a good "Group Theory" book

    I am physics student.I know basic definition of topological space.I want a book(or may be any web note or video lecture) where topology spaces of various groups are rigorously discussed.
  22. Dilatino

    Introduction to Young-tableaux and weight diagrams?

    I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods are applied in physics, such as in the context of the standard model and beyond for example. I am...
  23. Dilatino

    How can I construct the 4D real representation of SU(2)?

    An element of SU(2), such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as U(x) = e^{ixT_1} = \left( \begin{array}{cc} \cos\frac{x}{2} & i\sin\frac{x}{2} \\ i\sin\frac{x}{2} & \cos\frac{x}{2} \\ \end{array} \right) = \left(...
  24. applestrudle

    Group theory? This solution doesn't make sense....

    Case 2: I get that D = c I means A must also be proportional to I but how does that mean B must be diagonal? Question: Answers:
  25. S

    Good introductory textbooks for group theory

    Hi there. Can anybody recommend a good textbook for an undergraduate wanting to study group theory (especially representation theory). I'm thinking of reading "visual group theory" by Carter for conceptual understanding but I also need a book to study alongside this that gives a more formal...
  26. N

    Quantum Mechanics and Group Theory questions

    Hello all. I am new here. I am in the last quarter of a 3 quarter sequence of undergrad quantum mechanics and I just had some conceptual questions (nothing pertaining to homework). We just recently covered Berry's Phase and the Dynamical Phase. Now I wanted to start with a more basic quantum...
  27. HaLAA

    Show the group of units in Z_10 is a cyclic group of order 4

    Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework EquationsThe Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this...
  28. S

    Meaning of representations of groups in different dimensions

    Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...
  29. HaLAA

    Show (H,+) is isomorphic to (C,+)

    Homework Statement Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+) Homework Equations Isomorphism, Group Theory, Binary Operation The Attempt at a Solution Let a,b,c,d∈ℝ Define f : M→ℂ by f( (a -b) (b a) ) = a+bi 1-1: Suppose f( (a -b) (b a) )= f( (c...
  30. Futurestar33

    Comparing Group theory and Electromagnetism

    Homework Statement Good afternoon, How can you mathematicaly talk about how how group theory compares to electromagnetism. Homework Equations e^iθ=Cosθ+iSinθ The Attempt at a Solution I know that the above formula is because of a sin wave and a cosine wave. Put them together and you get a...
  31. Primroses

    Why are invariant tensors also Clebsch-Gordan coefficients?

    On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...
  32. T

    Abstract Algebra; Group Theory Question

    Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?
  33. c3po

    Finding the members of the Lie algebra of SO (n)

    Homework Statement Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression: R.RT= I with respect to the parameter t...
  34. S

    Intro Math Introductory Book on Group Theory for HS freshman?

    I am looking for an introductory book on group theory for my son who is a high school freshman. He has a good grasp of the basics of mathematics and is ready to take calculus classes. He has a very strong intuitive grasp of symmetry and transformations, so I thought that he may be ready to be...
  35. PsychonautQQ

    Group Theory: Element of Order 2 in Groups of Even Order

    Homework Statement If G is a group of even order, show that it has an element g not equal to the identity such that g^2 = 1. Homework Equations None The Attempt at a Solution What I wrote: If |G| = n, then g^n = 1 for some g in G. Thus, (g^(n/2))(g^(n/2)) = 1, so g^(n/2) is the element of...
  36. J

    What are some real-world applications of cyclic groups and Galois fields?

    Can anybody name some real world applications of group theory? I would be particularly interested to hear any uses of cyclic groups to solve everyday problems one could encounter.
  37. Abolaban

    Question on group theory: simplest math construction

    Hello Big Minds, In the following analysis...It is said that D_2 contains three subgroups Z_2...why did he choose a mathematical constuction contains only two of the the three subgroups? shouldn't he use the three in his construction? what will happen if he used the three? [from the book of...
  38. P

    MHB Exploring the Application of Maths in Music: Group Theory and Beyond

    Is there any application of maths in music, which topic is directly used. I've heard group theory is used, but how it is used. Plz help..i m going to work on a project that will show how maths works in music or sound system.
  39. A

    How to show G/Z(R(G)) is isomorphic to Aut(R(G))?

    I am working on this problem with lots and lots of nesting definitions like this following, and I have been trying to get help from here as well as http://www.quora.com/How-do-I-prove-G-Z-R-G-is-isomorphic-to-Aut-R-G , but none gave me complete help: Show that ##G/Z(R(G))## is isomorphic to a...
  40. M

    Why does the order of the center of a non-abelian p-group have to be p?

    Homework Statement Let G be a non-abelian group with order ##p^3##, p prime. Then show that the order of the center must be p. Homework Equations Theorem in our book says that for any p-group the center is non-trivial and it's order is divisible by p. Class eq. ##|G|= |Z(G)| + \sum{[G ...
  41. L

    Algebra Any recommendations for group theory books with applications in relativity?

    Hello, I’m looking for introductory books/notes on group theory and algebra. We are using “Groups” by Jordan and Jordan in class, but I am looking for something a little more in-depth. I wouldn’t mind a book that was entirely focused on the pure maths side but if it had lots of applications to...
  42. L

    DMRG. Density matrix renormalization group theory

    Why density matrix renormalization group theory works only for 1D systems?
  43. A

    Solving Simple Group Homomorphism Problem: Proving phi(G) is Subgroup of N

    I have this problem on simple group's homomorphism: Let ##G′## be a group and let ##\phi## be a homomorphism from ##G## to ##G′##. Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G′## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subset N##. And last year somebody...
  44. aabottom

    Good books on the group theory of quantum mechanics

    Hello I'm looking for good books on the group theory of quantum mechanics. I have a BS in Physics, MS in Electrical Engineering and decades of work experience in building lasers, and R&D in laser systems, optics & infrared sensing systems. My main goal is to study & understand quantum...
  45. A

    Validating Logic in a Group Theory Problem

    To make a very long story short, in a group theory problem I am working on, I need to prove this: ##A \lhd B \Rightarrow A'\neq A##, where ##A## and ##B## are finite and ##A'## is called the commutator subgroup: ##\begin{align} A' :&= [A, A] \\ &= \langle [x, y] \mid x, y \in A \rangle \\ &=...
  46. Prof. 27

    Unsure of how this is read

    Homework Statement Hi, the problem is, I'm unsure of how this is read. The context is that I'm currently reading on group homomorphisms and they use this notation. I think it is also used in composition of functions, but I'm unsure of how it is read there as well. h:X-->Y So how would I say...
  47. PsychonautQQ

    Group Theory Question: Ker(p) and Homomorphisms Explained in Detail

    Homework Statement Let p: G-->M be a group homomorphism with ker(p) = K. If a is an element of G, how that Ka = {g in G | p(g) = p(a)} Homework Equations none needed The Attempt at a Solution Okay, I've been struggling with this problem for awhile and I've ran into a problem: -Let g be an...
  48. A

    Automorphism Group of Radical of Finite Group

    I am working on a problem on automorphism group of radical of finite group like this one: Here are what I know and what I don't know: ##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the...
  49. A

    Subnormal p-Sylow Subgroup of Finite Group

    I am self-studying a class note on finite group and come across a problem like this: PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).## Where ##O_p(G)## is the subgroup generated by all subnormal p-subgroups of ##G##; ##E(G)## is the...
  50. T

    Isomorphism under differentiation

    Is 《sinx》under differentiation a valid cyclic group.
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