Group theory Definition and 365 Threads

Huzzah.

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...

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...

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...

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

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...

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...

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.

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...

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...

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

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...

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...

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...

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 =...

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...

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...

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...

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!

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

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...

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...

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"?

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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. *...

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...

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...

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...

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

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...

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?

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...

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. \[...

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...

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!

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?

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