Group theory Definition and 365 Threads

"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...

[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...

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...

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

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...

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?

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...

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...

[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...

[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...

[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...

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...

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...

[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...

[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.

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...

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...

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

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

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...

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...

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...

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

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

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?

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...

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!

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...

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...

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)...

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.

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...

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

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...

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 ?

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...

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

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...

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...

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...

Hi, I have the following problem : I generate GaAs (zinc blende structure) supercells, and then I want to replace some As atoms by N atoms. Let's say I have fcc conventional cell repeated twice in the x, y and z direction so that I have a total of 64 atoms, 32 of Ga and 32 of As. 8 atoms...

hi, can anybody recommend any good textbooks for group theory? I've just started a 3rd year course called groups and geometry and the lecture notes aren't the greatest of help. The lecturer just writes down some algebra regarding the topics without much notes to support them and doesn't...

It's always annoying when one finds in books (written by (theoretical) physicists for (theoretical) physics students) statements like those below without a mere cross-reference for a mathematically-rigurous proof. And that's what I'm searching for right now: either point me to some books, or...

I have the definition that if F is a finite field then a \in F is a primitive root if ord(a) = |F|-1. Now what I don't understand is how exactly are there \phi(|F|-1) primitive roots? (Note: This material is supposed not to use any group theory.)

Hi all. I'm studying Wybourne book on group theory. I didn't understand this expression: U(p,q) I know what U(p+q,C) and U(n) means, but I'm unfamiliar with the notation of the above statement. Thanks in advance. Somy:smile:

Group Theory, please help! Okay, so I'm stuck on a couple questions from my homework, and any guidance would be much appreciated. 1. Prove that if G is a finite group with an even number of elements, then there is an element x in G such that x is not the identity and x^2 = e. I know...

A little intro: This is for a whirlwind intro to Group Theory as part of another class (QFT) in which we are not proving anything, simply introducing definitions and theorems. We are not using a textbook, simply some notes the professor has written up for this intro and the notes are very...

I need some help with what should be a really simple group theory proof, but for some reason I'm hitting my head against a wall on this one. I seem to be missing something simple to get me to the next step. I'm learning this stuff in Swedish, so I'm not sure about all the English words, but...

From what I've read of his work on FLT, I think Edgar Escultura is brilliant. I wonder why he hasn't gotten the recognition he deserves. Does anyone know where I can find more articles by him?

I've always been fascinated by Rubik's cube. I have developed solutions for it and all the related cubes 2x2, 3x3, 4x4, 5x5. For me the cube it is to group theory (of a partcular type of group) what a slide rule is to real arithmetic. Even "laboratory" might not be too stong a label for...