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. M

    Does AB=G Imply AB=BA in Group Theory?

    If A and B are subgroups of G and AB=G does it follow that AB=BA? If yes, why?
  2. T

    Basic Group Theory: Proof <A,B>n<C>=<AuBnC>

    In a group G, is it true that <A,B>n<C>=<AuBnC> where A,B and C are sets in G? Where <D> denotes the smallest subgroup in G containing the set D. Proof If g is in <A,B> and g is in <C> then g is capable of being generated by elements in A or B and also elements in C. So g is generated by...
  3. L

    Where Does Group Theory Belong on Physics Forums?

    Hi, I'm not finding group theory listed in any of the other math forums on physicsforums.com... so I'm wondering if it belongs here in the General Math forum or in another?
  4. L

    Group Theory - Help in learning

    Hi, I'm going through a group theory text on my own, as it is not formally covered in my undergrad curriculum. I've had a good (multiple course) background in the basics, differential equations, linear algebra with Hilbert spaces, etc., in my undergrad coursework in my physics major. I'd like to...
  5. P

    Understanding Group Theory and Vector Rotation in 3D

    Homework Statement I'm trying to see the relation of the rotation of a vector in a plane to the generator of rotations... I want to see how e^{-i \theta J} the rotation representation gives you the same result as acting on any vector with the rotation matrix say with the z direction fixed. \[...
  6. J

    R. Slansky, Group theory for unified model building

    I'm a beginning QFT student, trying to slog my way through Slansky's http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVP-46SPHMC-94&_user=4422&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059600&_version=1&_urlVersion=0&_userid=4422&md5=c09eed961c9201786651592d71de94f3"... I...
  7. X

    What is the significance of parity in group theory?

    Parity vs. group theory? Parity is a special property in Quantum mechanics. I don't know whether it relates to group thery? Is it O(2), U(1), or others? Thank you!
  8. A

    Group Theory Notation: SO(n,1) & Poincare Groups Explained

    1. I understand the meaning of group SO(3) etc, but what is meant by say SO(n,1) or Poincare(n,1)group? 2. what is the importance of double cover of a group in physics?
  9. J

    Exploring the Difference Between C2 and C4 Operations in Group Theory

    What is the difference between the C2 and C4 operations on one hand for Oh and on the other, the 3C2=(C4)2?
  10. C

    Group theory and Spacetime symmetries

    "A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why? The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition...
  11. E

    Solved: Group Theory Problem: Cyclic Group of Order 15

    [SOLVED] group theory problem Homework Statement A cyclic group of order 15 has an element x such that the set {x^3,x^5,x^9} has exactly two elements. The number of elements in the set {x^{13n} : n is a positive integer} is a)3 b)5 Homework Equations The Attempt at a Solution...
  12. C

    Group Theory, permutations formula

    When proving that A_n with n \geq 5 is simple, we require the following lemma: If N is a normal subgroup of A_n with n \geq 5 and N contains a 3-cycle, then N = A_n. The proof is actually given for us in the lecture notes, however he utilizes a formula that I'm not sure how he derived: Let f...
  13. T

    Who introduced group theory into quantum field theory?

    I am trying to search for the first person who introduced group theory into quantum field theory.
  14. O

    What is group theory all about?

    A short answer to this question is perhaps "symmetry", or one might specify the properties that define a group (closure, associativity etc.). But are there less precise and more general ways of describing group theory that someone could point me to --- ways that perhaps better express the...
  15. R

    Exploring the Importance of Symmetry in Group Theory

    I'm not an expert in Abstract Algebra, I am mainly an analyst. Is there anyone versed in Group Theory that can kind of discuss the theory and it's ramifications?
  16. E

    How Does Group Theory Aid in Predicting Vibrational Modes in Spectroscopy?

    Hi Everyone, I am giving a talk on group theory as related to spectroscopy (IR & Raman) and I was curious about how to explain the jump from knowing which modes in the charcter table are Raman active to predicting what kind of vibration they will have. For example the Td point group has raman...
  17. E

    Group Theory Question: Is (Left) Multiplication by g an Isomorphism in G?

    Homework Statement true or false: If G is a group and g is in G. Then (left) multiplication by g is an isomorphism from G to GHomework Equations The Attempt at a Solution I am pretty sure it is true since ax=b always has a solution if a and b are in group. But can someone just confirm this...
  18. E

    Solved: Find Solutions for Group Theory Problem in Z_12

    [SOLVED] group theory problem Homework Statement Find all solutions of the equation x^3-2x^2-3x=0 in Z_12. Homework Equations The Attempt at a Solution We first factor the polynomial into x(x-3)(x+1)=0. Recall that Z_12 is not an integral domain since 12 is not prime (e.g...
  19. E

    Solve Group Theory Problem: (Z_4 x Z_4 x Z_8)/<(1,2,4)>

    [SOLVED] group theory problem Homework Statement Classify the factor group (Z_4 cross Z_4 cross Z_8)/<(1,2,4)> according to the fundamental theorem of finitely generated abelian groups. Homework Equations The Attempt at a Solution <(1,2,4)> has order 4 so the factor group has...
  20. E

    Proving Normality of the Commutator Subgroup in Group Theory [SOLVED]

    [SOLVED] group theory question Homework Statement Prove that the commutator subgroup is normal.Homework Equations The Attempt at a Solution Let H be the subgroup generated by all of the commutators. We want to show that H is normal. Let x be in yH. Then, y^{-1}x=aba^{-1}b^{-1} for some a,b...
  21. M

    More group theory (or module theory)

    Homework Statement Given an abelian group G with generators x and y, and relations 30x + 105y = 42x + 70y = 0, show it's cyclic and give its order. Homework Equations The Attempt at a Solution I'm guessing the proof basically involves cleverly adding 0 to 0 to show that x = ry or...
  22. E

    Proving Torsion Subgroup Normality and Torsion-Free Property in Abelian Group G

    Homework Statement Prove that the torsion subgroup T of an abelian group G is a normal subgroup of G, and that G/T is torsion free.Homework Equations The Attempt at a Solution The second part of this exercise makes absolutely no sense to me. We know nothing about G, so why is there any reason...
  23. E

    Proving G/H is Abelian: Normal Subgroups in Abelian Groups

    [SOLVED] group theory question Homework Statement A student is asked to show that if H is a normal subgroup of an abelian group G, then G/H is abelian. THe student's proof starts as follows: "We must show that G/H is abelian. Let a and b be two elements of G/H." Why does the instructor...
  24. E

    Yes, it should be |G'|. Thank you for catching that!

    [SOLVED] group theory Homework Statement Let \phi:G \to G' be a group homomorphism. Show that if |G| is finite, then |\phi(G)| is finite and is a divisor of |G|.Homework Equations The Attempt at a Solution Should the last word be |G'|? Then it would follow from Lagrange's Theorem.
  25. E

    Group Theory: Isomorphism between Z_3 x Z_4 & Z_12

    [SOLVED] group theory Homework Statement My book says that Z_3 cross Z_4 is isomorphic to Z_12, which I am confused about because Z_3 cross Z_4 has four different generators and Z_12 only has 1. EDIT: wait that is not true, Z_12 has generators 1,5,7,11 It is probably true in general that the...
  26. quasar987

    Exploring Abelian Groups and Normal Subgroups in Group Theory

    Homework Statement G is abelian, A is normal in G, B is a subgroup, a1, a2 in A, b1,b2 in B, c_g denotes the congugation by g automorphism. why must a_1c_{b_1}(a_2) = a_2c_{b_2}(a_1) imply that c_{b_1}(a_2)=a_2 and c_{b_2}(a_1)=a_1 The Attempt at a Solution In other words, why couldn't...
  27. C

    Ring Theory and Group Theory questions

    Hey everyone, I was hoping to grab some quick advice on these two topics. Specifically, I'm a 4th year physics undergrad with all the standard physics and math courses, as well as real analysis up to lebesgue measure theory/integration theory+hilbert spaces,etc., and grad level PDEs. I...
  28. P

    What is the significance of SU(2,4) in Group Theory?

    What does this expression, SU(2,4), mean?
  29. E

    Group Theory Question: The Symmetry of Right and Left Multiplication in Groups

    Homework Statement If G is a group, is it true that right multiplication by a given element is a homomorphism but left multiplication is not? That does not really make sense to me because aren't right and left multiplication "symmetric"? Homework Equations The Attempt at a Solution
  30. T

    Group theory question

    Homework Statement If p is a prime and G is group of order p^2, then show that G is abelian. Homework Equations n/a The Attempt at a Solution I first consider Z(G), the centre of G. Since it is a normal subgroup of G, then by Lagrange's Theorem, |Z(G)| divides |G|. Hence |Z(G)| = 1, p or...
  31. JasonRox

    Solve Group Theory Problem: Proving HuX is G

    I'm stuck on this one... I'm studying for my midterm so I'm solving problems for practice. Here's one of them... Let H be a normal subgroup in G, and let v be the natural map from G to G/H, and let X be a subset of G such that the subgroup generated by v(X) is G/H. Prove that the subgroup...
  32. Z

    What Is the Best Order to Study Group Theory, Tensor Analysis, and QFT?

    I want to study 3 subjects on my own,the subjects are Group Theory, tensor analysis, and QFT. I know this might be a silly question, but regardless of what textbook material i have or how much I know, what is the best order to study these 3 subjects ? I feel I should leave QFT to the last...
  33. T

    Group Theory - Sylow's Theorem

    Hey ho, just wondering if anyone knows of any decent sites which deal with Sylow's theorem. Cheers, Twiddles :)
  34. C

    Group Theory Q: Deriving (10.80) Detail Calc

    See the attachment(from Quantum Mechanics-Symmetries 2nd W.Greiner B.Muller1994W-p345), how to derive (10.80) please show me the detail calculation
  35. C

    What do SO(N) and U(N) mean in group theory?

    what do these groups mean? I think that SO(N) means rotation, but I am not sure? Also does the N mean the number of dimensions?
  36. S

    Help Group Theory Question(s)

    Homework Statement Take G to be the cyclic group with 12 elements. Find an element g in G such that the equation x^2 = g has no solution. Homework Equations Notation: Z = set of integers. A group is said to be commutative or Abelian if the operation * satisfies the commutative law...
  37. D

    Group theory and characters table

    Hello, where can I find something about Group Theory and Characters Table? How to construct it, what can we do with it, etc... Thank you!
  38. M

    Group Theory + Fermat's Little Number Theorem

    This isn't a homework problem as it is me working through some math texts in order to add some "rigor" to my physics/engineering acumen. As an undergrad, I attempted to take a course in abstract algebra, but had to drop it due to scheduling problems. I've been working through my copy of...
  39. A

    What Determines the Isomorphism of G/X in Group Theory?

    Hi, Consider the simply connected group G of all 3 by 3 matrices [1 a b 0 1 c 0 0 1 ] where a,b,c are in C. The center of G is the subgroup Z(G)={ [1 0 b 0 1 0 0 0 1] ; b is in C} So Z(G) is isomorphic to C and therefore the discrete subgroups of Z(G) are just lattices X of rank 1...
  40. A

    Group Theory: Proving Non-Isomorphism of G_n and G_m

    Hi all, Consider the groups G_n= {(a,b) in (C*) * C ; (a,b).(a',b')=( aa', b+(a^n)b' )} where n is in N. Show that if n and m are distinct, then the two groups G_n and G_m are not isomorphic. Ps. In fact G_n = G/ nZ , where G is the group of pairs (t,s) in C * C with group law (t,s)...
  41. M

    Proving a = a^-1 implies a^2 = e in group theory

    Let G be a group. Let 'a' be an element of G. Let e be the identity of G Prove that if a = a^-1 then a^2= e. Is the proof below correct ? Suppose a = a^-1. Then a^2 = aa = a(a^-1) = e.
  42. I

    Finding Surjective Homomorphisms from Symmetric Groups to Cyclic Groups

    Hello, I am having difficulty with the following problem in Group theory: How do you positive integers r such that there is a surjective homomorphism from S_n (This is the symmetric group of order n) to C_r (This is the cyclic group order r) for some n ? I am not sure where to even start...
  43. M

    Proving Commutativity in Groups with a^2 = e for all a in G

    Could you also help me start this question show that if a^2 = e for all a in G then G must be commutative. (where e is the identity) thanks
  44. M

    Is Every Group with Only Cyclic Subgroups Itself Cyclic?

    I m having trouble with a couple group theory proofs. I just have no clue how to start. If u could put me on the right path that would be great. first prove of disprove that if every subgroup of a group G is cyclic, then G is cyclic. and second prove or disprove that every group X of...
  45. M

    Group Theory: Necessary & Sufficient Conditions for G_W = G_(W)

    Let n be in |N. Let G denote S_n , the symmetric group on n symbols. Let W be a subset of {1, 2, ..., n}. Write down VERY simple necessary and sufficient conditions on |W|, for G_W to equal G_(W). We know G_W < G_(W) < G , but now what ?
  46. S

    Group Theory - Isomorphism question

    GroupTheory - Isomorphisms Hey I'm stuck on these 2 questions, was wondering if anyone could assist me: Let G be a nontrivial group. 1) Show that if any nontrivial subgroup of G coincides with G then G is isomorphic to C_p, where p is prime. (C_p is the cyclic group of order p!) 2) Show...
  47. N

    Group Theory & Physics: Physicist's Textbook

    What is the physicists must have textbook on group theory and its applications to physics?
  48. M

    Group Theory - Prime Order + Subgroups

    Problem: "Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime. (Do not assume at the outset that G is finite)." Basically, I'm pretty sure I can do this problem. I'm just unsure of how to prove that all infinite groups...
  49. A

    How to Solve Group Theory Problems and Understand Underlying Concepts?

    1. (a) Find the number of p-Sylow subgroups in G = GL(n,Fp). (b) For two p-Sylow subgroups H and J of G, consider the number of elements in the intersection. What number appears this way for given p and n? (c) For a given p-Sylow subgroup H of G, find the number of p-Sylow subgroups with the...
  50. E

    Permutation \psi in S_{10} of Order 20: Odd or Even?

    Here's another question I'm lost on: "Show that the permutation \psi contained in S_{10} of order 20 is odd" I know that the order of the permutation is the least positive integer n such that \psi^n is the identity permutation. I don't know how to check whether the permutation is...
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