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. B

    Short problem on group theory q.1

    short problem on group theory q.1[urgent due in 13 hrs] due now
  2. B

    Is G/N Abelian If N Contains All Commutators in a Group?

    urgent another group theory problem sorry Homework Statement Let G be a group with normal subgroup N. Prove that G/N is an abelian group of and only of N contains elements aba^{-1}b^{-1} for all a,b in G. Homework Equations commutator The Attempt at a Solution G/N i know it is the...
  3. B

    Interesting group theory problem

    Homework Statement ~ Homework Equations center The Attempt at a Solution i can do part a. can you give me hints on part b and c?
  4. B

    Group theory problem exceedinly difficult and no one can solve it. can you?

    Homework Statement ~p.s. it should be H/N is abelian, not H being abelian.Homework Equations subgroup The Attempt at a Solution for a) i have some idea for b) i have no idea. help~ :)
  5. J

    A very challenging question regarding in basic algebra group theory?

    1.Why Aut(G)=S_G implies G is trivial? I search through the internet and no answer.2.Here is another very difficult conception question which has different answers from my professor and wikipedia: Difference between Symmetry group,automorphism group and Permutation group? From...
  6. A

    Meaning of colon in group theory, if not subgroup index?

    I am reading a paper where the author uses colons in the description of groups. Example (not verbatim): "This subgroup is isomorphic to (Z_5 X A_4):Z_2". Several subgroups are described in the same way (as (G_1 x G_2):G_3) throughout the paper. I have seen the colon in G:H to indicate the...
  7. M

    Unlocking the Power of Group Theory: A Guide to Success

    Huzzah.
  8. P

    Group Theory Question involving nonabelian simple groups and cyclic groups

    Homework Statement Let A be a normal subgroup of a group G, with A cyclic and G/A nonabelian simple. Prove that Z(G)= AHomework Equations Z(G) = A <=> CG(G) = A = {a in G: ag = ga for all g in G} My professor's hint was "what is G/CG(A)?" The Attempt at a Solution A is cyclic => A is...
  9. E

    Very Elementary Group Theory Problem

    Hi. I'm new to Group Theory and wanted to see if I had the right train of thought for this problem. Homework Statement Let S be a set with an associative law of composition and with an identity element. Prove that the subset of S consisting of invertible elements is a group. Homework...
  10. N

    A book introducing Quamtum Field Theory from a group theory approach

    I'm looking for a book that describes Quantum Field Theory from a group theory approach for mathematical physicists (with emphasis on the physics part). Ideally I want it to first describe and define groups, representations and irreducible representations. The more rigorous the math, the better...
  11. P

    Applications of group theory

    i just want to know few practical applications of group theory...please help!
  12. L

    What are the properties of normal subgroups in groups of prime order?

    Hi, I've been vanquished by probably easy problems once again. Homework Statement 1. Let G be a group of order p^2 (p prime number), and H its subgroup of order p. Show that H is normal. Prove G must be abelian. 2. If a group G has exactly one subgroup H of order k, prove H is normal in...
  13. Z

    Learning Point Group Theory: Challenges of Lee Groups

    Hi Everyone, Back in college i informally learned what i would call point group theory. Most of it never touched on continuous transformations. When I learned it back then it was all pretty straight forward. Recently I have been trying to learn about Lee groups (to understand symmetries in...
  14. R

    Group theory, subgroup question

    Let A be a subgroup of G. If g \in G, prove that the set {g^{-1} ag ; a \in A} is also a subgroup of G. Thanks for any help.
  15. R

    Group Theory, cyclic group proof

    Homework Statement Prove that Z sub n is cyclic. (I can't find the subscript, but it should be the set of all integers, subscript n.)Homework EquationsLet (G,*) be a group. A group G is cyclic if there exists an element x in G such that G = {(x^n); n exists in Z.} (Z is the set of all...
  16. T

    Problem understanding Group Theory question

    Hello all, my first post, hope to be a regular forum goer. Any help understanding this problem would be appreciated. Homework Statement "Consider the following functions: f(x) = 1/x ; g(x) = 1/(1-x) defined on the set R\{0,1} = (-∞,0) U (0,1) U (1,∞) How many total functions can be...
  17. K

    One group modified and Two group theory

    What is the big difference between the two reactor calculations. They seem to be virtually the same?
  18. K

    Group theory: Conjugates, commutativity

    Homework Statement In S8, consider p= (1 3)(2 4 5 6) and q=(1 3)(2 4)(7 6 5) 1. Find the number of conjugates of p and the number of conjugates of q 2. Find the number of permutations that commute with p and the number that commute with q Homework Equations The Attempt at a...
  19. O

    Abstract Alg- Group theory and isomorphic sets.

    Homework Statement I am suppose to determine if the following list of groups are isomorphic and if they are define an isomorphic function for them. a. [5Z, +],[12Z, +] where nZ = {nz | z\inZ} b. [Z6, +6]], [S6, \circ] c. [Z2, +2]], [S2, \circ] Homework Equations +6 means x +6] y = the...
  20. U

    Relationship between group theory and particle physics?

    I'm just a year 12 student with an interest in mathematics and physics, but I have a question (or rather a few) for particle physicists/mathematicians out there. I have read a little about abstract algebra - to about the extent of knowing the definition of a group, its relationship to...
  21. A

    Group theory, is my solution correct?

    Homework Statement if H is a normal subgroup of G and has index n, show that g^n is in H for all g in G. The Attempt at a Solution Take H a normal subgroup of a group G. Take g in G. Consider gH in the quotient group G/H. Because |G/H| = [G:H] = n, (gH)^n = eH. But g^nH =...
  22. F

    Proof on Normal Subgroups and Cosets in Group Theory

    This is a proof I am struggling on ... Let H be a subgroup of the permutation of n and let A equal the intersection of H and the alternating group of permutation n. Prove that if A is not equal to H, than A is a normal subgroup of H having index two in H. My professor gave me the hint to...
  23. S

    Group Theory Help: Show (x*y*z^-1)^-1 = x*y^-1*z^-1

    Homework Statement Let G(*) be a group. If x.y are elements of G show that (x*y*z^-1)^-1 = x*y^-1*x^-1 Homework Equations The Attempt at a Solution I first took the left side of the equation and computed the inverse and I got x^-1*y^-1*z I then let this equal to the righthand...
  24. 3

    Question about Lagranges theorem in Group Theory

    Homework Statement If H is a subgroup of a finite group G, and if the order of G is m times the order of H, |G|=m|H|, adapt the proof of Lagrange's theorem to show that gm! is an element of H for all g in G. The Attempt at a Solution My thoughts so far were to think that we can divide G...
  25. M

    Group Theory Permutation (Hints and )

    Homework Statement 1. Let n ≥ 2. Let H = {σ ∈ S_n: ord(σ) = 2}. Decide whether or not H is a subgroup of S_n. 2. Let G be a group of even order. Show that the cardinality of the set of elements of G that have order 2 is odd. The Attempt at a Solution 1. I have no idea where to start with...
  26. S

    Which book is well suitable for the study of tensor analysis and group theory

    Please introduce me a good book for my self_study of tensor analysis and group theory. I am a sophermore preparing to self-study them!
  27. M

    Group theory, Lagrange theorem

    Dear all, The question I've been struggling with is supposed to be solved using the way Lagrange's thm was proven( with number of cosets and stuff). However, it remains a mystery how to do it: Let G be a finite group and H<G with |G|=m|H|. Proof that g^{m!} \in H, \forall g \in G
  28. E

    Solve Group Theory Problem - Prime Order of G must be p^n

    Hi I have a problem I just can't seem to solve, even though the solution shouldn't be too hard Let G be a finite abelian group and let p be a prime. Suppose that any non-trivial element g in G has order p. Show that the order of G must be p^n for some positive integer n. Anyone got any...
  29. M

    Proving Isomorphism Between Finite Group and Z_2 Using Group Theory

    Let G be a finite group. For all elements of G (the following holds: g^2=e(the idendity.) So , all except the idendity have order two. Proof that G is isomorphic to a finite number of copies of Z_2 ( the group of adittion mod 2, Z_2 has only two elements (zero and one).) I can try to tell...
  30. D

    Finding Probability of Generating Symmetric Group Sn

    This is from Wikipedia's "unsolved problems in math" section: Finding a formula for the probability that two elements chosen at random generate the symmetric group Sn Can someone explain that to me? I know what Sn is. What are these two "elements"?
  31. P

    What is the inner product of a linear operator on a complex inner product space?

    Homework Statement Let T : V \rightarrow V be a linear operator on a complex inner product space V , and let S = I + T^{*}T, where I : V \rightarrow V is the identity. (a) Write <Sx,x> in terms of x and Tx. (b) Prove that every eigenvalue \lambda of S is real and satisfies \lambda\geq 1. (c)...
  32. R

    Group Theory and Social Groups.

    Hi, I have a question related to Group Theory and its interpretation from a social point of view. if we suppose, that a group of Humans can be considered as an algebraic structure : a group (G,◦) with a set of elements and a set of axioms like closure, associativity, identity and...
  33. K

    Energy levels and Group theory

    Homework Statement Calculate the energy of the excited states of neon that is obtained by promoting one of the 2p electrons to the 3p shell. Use LS coupling, neglect spin-orbit interaction.Homework Equations The ground state cofiguration of Neon is 1s2, 2s2, 2p6. The Attempt at a Solution I...
  34. M

    Unlocking the Weinberg-Salam Theory: A Guide to Group Theory Literature Online

    Hi all! I want to learn the Weinberg-Salam theory of weak and el.mag. interactions and for that a good knowledge of group theory is required. Can someone advise a good book from internet about Group Theory? Any help will be appreciated. P.S. this is my first post on this site, I hope I put...
  35. A

    Group Theory - specific non-abelian case

    Homework Statement Let G be an Abelian group and let H+{x^3 : x is an element of G} Find a non-Abelian group in which H is not a subgroup Homework Equations I wish it was that easy... The Attempt at a Solution I looked at the quaternion group, and some other matrix groups, but...
  36. Z

    Additional Group Theory Issues

    I really don't get this group theory stuff at all. These should be simple questions, but alas not... Homework Statement Assume that * is an associative operation on S and that a is an element of S. Let C(a) = {x: x is an element of S and a*x = x*a} Prove that C(a) is closed with...
  37. Z

    Group Theory Sets and Mappings

    Homework Statement Prove that f: S -> T is one-to-one if and only if f(AnB) = f(A) n f(B) for every pair of subsets A and B of S Homework Equations See above The Attempt at a Solution Part 1: Starting with the assumption f(AnB) = f(A) n f(B) Let f(a) = f(b) [I'm going to...
  38. Z

    How to Prove Stabilizers in Group Theory Using X and G?

    Suppose that G acts on the set X. Prove that if g \in G, x \in X then StabG(g(x)) = g StabG(x) g-1. Note: g StabG(x) g-1 by definition is {ghg-1 : h \in StabG(x)} My attempt at the problem is: Let a \in StabG(g(x)), then a(g(x)) = g(x) by definition. Also Let b\in StabG(x), then b(x) = x...
  39. W

    Group Theory Hints: Proving a Subgroup of Order 2n+1 for |G| = 4n+2

    Hi everyone This is the same question as was asked about in this topic, but I can't post in that one (presumably because it's archived?) Homework Statement Suppose G is a group with |G| = 4n+2. Show that there is a subgroup H < G such that |H| = 2n + 1. Use Cauchy's theorem, Cayley's...
  40. B

    Solving Group Theory Problems: Sylow, Abelian, and Order 36

    1. Let G be a fintie group whose order is divisible by a prime p. Assume that (ab)^p = a^p.b^p for all a,b in G. Show that the p-Sylow subgruop of G is normal in G. 2. Find the number of Abelian groups of order 432. 3. Let G be a group of order 36 with a subgroup H of order 9. Show that H...
  41. G

    Group Theory (Physics): Is It Worth It for Physicists?

    Hello there,i'm at my last semester in physics undergrad.I wanted to get group theory last semester but it I was already full with other subjects and research,so I went this semsester and took the group theory taught in the math department.Well,at first I was totally lost (me and the 2 best math...
  42. MathematicalPhysicist

    Exploring Isomorphism in Groups of Order 4

    The question: Prove that each group of order 4 is isomorphic to Z/4Z or the Klein Group: (Z/2Z)x(Z/2Z). Attempt at solution: basically I think that a group of order 4 has e,a,b,c then this group can be characterise by the ordering 0,1,2,3 in the group Z/4Z or (0,0),(0,1),(1,0),(1,1) where...
  43. J

    Group theory : inverse of a map.

    Homework Statement Let H, K be subgroups of a finite group G. Consider the map, f : H \times K \rightarrow HK : (h,k)\rightarrow hk. Describe f^{-1}(hk) in terms of h, k and the elements of H\cap K. Homework Equations HK = \{hk : h \in H, k \in K \} f^{-1}(hk)=\{ (h',k') : f(h',k')=hk...
  44. S

    Question about Lie Brackets in Group Theory

    What does it mean when a Lie Bracket has a subscript + or - directly after it? I found this notation in http://en.wikipedia.org/wiki/Special_unitary_group" under the fundamental representation heading Those are Lie Brackets, right? I know Lie Brackets are being used elsewhere in the article.
  45. T

    Is A3 Commutative and Cyclic?

    Homework Statement Every nontrivial subgroup H of the symmetric group with 9 elements containing some odd permutation contains a transposition. It does seem the case that if a subgroup of H of the symmetric group with 9 elements contain an odd permutation then certainly a transposition...
  46. K

    Identity element for group theory

    Homework Statement Show that (S, *) is a group where S is the set of all real numbers except for -1. Define * on S by a*b=a+b+ab The Attempt at a Solution Well I know that i have to follow the axioms to prove this. So I started with G1 which is associativity. This one I got to...
  47. D

    Introduction to Group Theory - Abstract Algebra

    Homework Statement Prove that if (ab)2 = a2b2 in a group G, then ab = ba.Homework Equations * For each element a in G, there is an element b in G (called the inverse of a) such that ab = ba = e (the identity). * For each element in G, there is a unique element b in G such that ab = ba = e. *...
  48. J

    Does group theory deal with asymmetry?

    I have a question from which you should notice that I do not have much of a clue abot group theory. At least not yet. The question is about that many introductory articles about group theory seem to refer to the use of group theory with rotations of bodies and their related symmetry. What...
  49. C

    Group Theory for Unified Model Building

    Hi everyone, I want to ask everybody if someone knows a book, or some lecture notes available on the net, to lear how to decompose the Lie Groups in irreps in physical notation, like 8_v \otimes 8_v=1+28+35 that can be found everywhere on books like BB&S or Polchinski. It is really hard...
  50. H

    Exploring Group Theory: 10 Non-Isomorphic Groups with Orders 25-29

    Dear All, Is it true that one can find some 10 groups (from different isomorphism classes) with order between (and including) 25 and 29 such that each pair of the same order are not isomorphic to each other? If so, how does one go about generating such a list and showing they are not...
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