Groups Definition and 906 Threads

I know that gauging a lie-goup with a kinetic term of the form: \begin{equation} \Tr{F^{\mu \nu} F_{\mu \nu} } \end{equation} Is not allowed for a non-compact lie group because it does not lead to a positive definite Hamiltonian. I was wondering if anyone knew of a general way to gauge...

Let p be a prime. Let H_{i}, i=1,...,n be normal subgroups of a finite group G. I want to prove the following: If G/H_{i}, i=1,...,n are abelian groups of exponent dividing p-1, then G/N is abelian group of exponent dividing p-1 where N=\bigcap H_{i} ,i=1,...,n. Proof: Since G/H_{i}...

Hello, I have a doubt on the definition of Lie groups that I would like to clarify. Let's have the set of functions G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}, that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the...

Hello, let's suppose I have the following system of curvilinear coordinates in ℝ2: x(u,v) = u y(u,v) = v + e^u where one arbitrary coordinate line C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2} represents the orbit of some point in ℝ2 under the action of a Lie group. Now I consider...

Hi there, I'm trying to figure out this question: Let A=[aij] be a 3x3 matrix with integer entries and let B=[bij] be it’s transpose. Let P and Q be the Abelian groups represented by A and B respectively. Show that P and Q are isomorphic by comparing the effects of row and column operations...

Hello, Some people on PF are currently self-studying calculus and topology. So we thought we might make a post here so that interested people could join us. We are doing the following books: Book of Proof by Hammack (freely available on http://www.people.vcu.edu/~rhammack/BookOfProof/)...

I am doing a survey of questions grouped into categories. Each question has a weight applied to it. I want to then total and average each category. Lastly, I want to total and average all the categories together. Here's the challenge: I want all of categories and the total average to have the...

My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B). Now however, the author I'm reading defines the group product multiplication as: (a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B. Does this...

I'm curious as to what all the science lovers on physics forums like to listen to. Feel free to throw in whatever styles or genres you like as well.

I know that this is a lot, but I would love some help. My trouble is at the end of Part II. Theorem (Cauchy’s Theorem): Let G be a finite group, and let p be a prime divisor of the order of G, then G has element of order p. Proof: Suppose G is a finite group. Let the order of G be k. Let p...

In alkenes, the more alkyl groups you're attached to means you are more energetically stable, and that we know the reactivity increases as the stability of the intermediate carbocation increases (with tertiary the most stable) I'm puzzled by this relationship, how is it that when it is more...

Proposition: If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order. Proof: Let G be a group with normal subgroup H. Suppose that every element of G/H has finite order and that every element of H has finite order. We wish to...

I was just wondering if anyone has ever tried this? Anyway, Over the summer, I'm taking calculus 2, multi variable calculus and university physics. This fall I will be in modern physics, university physics 2, diff equations and chemistry. I was just wondering if anyone would like to...

How can you tell if two Lie groups are isomorphic to each other? If you have a set of generators, Ti, then you can perform a linear transformation: T'i=aijTj and these new generators T' will have different structure constants than T. Isn't it possible to always find a linear...

Proposition: If G= <a> and b ϵ G, then the order of b is a factor of the order of a. Proof: Let G be a group generated by a. That is, G=<a>. Let b ϵ G. Since G is cyclic, the element b can be written as some power of a. That is, b=ak for some integer k. Suppose the order of a is n. Hence...

Hi, I wanted to see what people think about my current viewpoint on recognizing structures in abstract algebra. You count the number of sets, and the number of operations for each set. You can also think about action by scalar or basis vectors. So monoids groups and rings have one set...

Attached is my attempt at a proof. Please critque! :shy: Thank you!

I was reading on wikipedia on direct product of groups because I wanted find out if every subgroup of G \times H is realized as a direct product of subgroups of G and H. Apparently it is not, because the diagonal subgroup in G \times G disproves this. I'm a little confused, because I thought...

Show that in a finite cyclic group G of order n, writtten multiplicatively, the equation xm = e has exactly m solutions x in G for each positive integers m that divides n. Attempt... Proof: Let G be a finite cyclic group of order n, and suppose m is a positive integer that divides n. Let x be...

The question states: "Let G be a group and let Gn={gn|g ε G}. Under what hypothesis about G can we show that Gn is a subgroup of G?The set Gn is taking each element of G and raising it to a fixed number. I started my investigation by examining what happens if I take n=3 and considering the...

Hi, I will be writing my bechelors thesis on the application of renormalization groups and p-adic analysis on stock markets. The problem is that I don't understand it yet. I need advice on the best path how to learn it. What I know already: - I've read most of Mac Lanes Algebra, so I know...

Is it true that if there is monomorphism from group A to group B and monomorphism from group B to group A than A and B are isomorphic? i need some explanation. thx

Homework Statement find the functional groups in the following compound: Homework Equations C8H16O4The Attempt at a Solution I know there is an ether but there is also something else. What is it? I have tried to find a group that is in the compound, but I have had no luck. It seems to be only...

I've done some searching and have thus far come up empty handed, so I'm hoping that someone here knows something that I don't. I'm wondering if there has been any work on the enumeration of groups of order n (up to isomorphism); specifically, has anyone derived a generating function? Ideally...

how would you estimate number of groups of order 24? i do not need exact number, it is 15. I know that there are 5 groups of order 8 and there is 1 , 4 or maybe 7 sylow 3-groups. but i do not know what to do next

proove that if G and H are divisible groups and there is monomorphisms from G to H and from H to G than G and H are isomorphic

characterize primes p and q for which each group of order pq is nilpotent

There is a theorem for finite groups of isometries in a plane which says that there is a point in the plane fixed by every element in the group (theorem 6.4.7 in Algebra - M Artin). While the proof itself is fairly simple to understand, there is an unstated belief that this is the only point...

good day! i need to prove that the alternating group An is a normal subgroup of symmetric group, Sn, and i just want to know if my proving is correct. we know that normal subgroup is subgroup where the right and left cosets coincides. but i got this equivalent definition of normal group from...

Homework Statement Let a = (a1a2..ak) and b = (c1c2..ck) be disjoint cycles in Sn. Prove that ab = ba. The Attempt at a Solution Sn consists of the permutations of the elements of T where T = {1,2,3,...,n} so assume we take an i from T. Then either i is in a, i is in b, or i is in...

If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23 end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n. How would I then write this as Cn1 x Cn2...

If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23 end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n. How would I then write this as Cn1 x Cn2...

Hey guys, I'm having an issue with a question, namely Let x be a subset of S4. Is x a group? x = {e, (123), (132), (12)(34)} I don't really understand how I can test the 4 axioms of a group and how x being a subset of S4 would help?

Homework Statement Prove that no group of order 96 is simple. Homework Equations The sylow theorems The Attempt at a Solution 96 = 2^5*3. Using the third Sylow theorem, I know that n_2 = 1 or 3 and n_3 = 1 or 16. I need to show that either n_2 = 1 or n_3 = 1, but I am unsure how to do...

Homework Statement Show that Z18/M isomorphic to Z6 where m is the cyclic subgroup <6> operation is addition The Attempt at a Solution M = <6> , so M = {6, 12, 0} I figured I could show that Z18/M has 6 distinct right cosets if I wanted to do M + 0 = {6, 12, 0} M + 1 = {7, 13, 1}...

hey guys, I just want grasp the whole concept of quotient groups, I understand say, D8/K where K={1,a2} I can see the quotient group pretty clearly without much trouble however I start to get stuck when working with larger groups, say S4 For instance S4/L where L is the...

Hi I am trying to learn about quotient groups to fill the gaps on things I didn't quite understand from undergrad. Anyway I have a question regarding this: Can someone please explain how { 0, 2 }+{ 1, 3 }={ 1, 3 } in Z4/{ 0, 2 }? I would think since 0 + 1 = 1 and 2 + 3 = 1 under mod 4...

How to find the center of groups of order 8?

Consider field extensions of the form Q(u) where Q is the field of rational numbers and u=e^{\frac{2\pi i}{n}}, the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else...

hey, if you have the group S3 and treat it also as a g-set under the action x*g = g-1xg is it correct that the stabilizer of the g-set is just the identity element? thanks

I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups On page 6 (see attachment - pages 5 -6 Grove and Benson) we find the following statement: "It is easy to verify (Exercise 2.1) that the vector x_1 = (cos \ \theta /2, sin \ \theta /2 )...

If two elements in a group operate together and can create more then 1 answer (this answer is still a part of set, not foreign) is it still a group, if so why?

I would like to ask if somebody can verify the solution I wrote up to an exercise in my book. It's kind of long, but I have no one else to check it for me :) Homework Statement If H is a maximal proper subgroup of a finite solvable group G, then [G:H] is a prime power.Homework Equations Lemma...

I have a question regarding terminology here. The assignment is somewhat as follows: "If you think any of the following is a group, classify it along the following lines: finite, infinite discrete, finite-dimensional continuous, infinite-dimensional continuous." The definition of finite is...

I am reading Kane's book on Reflections and Reflection Groups and am having difficulties with some basic geometric notions - see attachment Kane paes 6-7. On page 6 in section 1-1 Reflections and Reflection Groups (see attachment) we read: " We can define reflections either with respect to...

"determine all abelian groups of order 1000. which of them contains exactly 3 elements of order two and which of them contain exactly 124 elements of order five?" ok now I have all the terminology down for this. But its the first time attempting such a type of question. How do I go about this...

I am reading Kane - Reflection Groups and Invariant Theory and need help with two of the properties of reflections stated on page 7 (see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7) On page 6 Kane mentions he is working in \ell dimensional Euclidean space ie...

Homework Statement According to wikipedia, one of the requirements of group is: For all a, b in G, the result of the operation, a • b, is also in G. So say we have 2 (2x2) matricies as elements of a group: \frac{0|1}{1|0} and \frac{w|0}{0|w^{2}} and the product \frac{0|1}{1|0} •...

Homework Statement For each m >= 2, find a group with a composition series of length 1 with a subgroup of length m. Homework Equations Simple groups iff length 1. If G is abelian of order p1^k1...pr^kr, then length G = k1 + ... + kr If G has a composition series and K is normal...

Hello, from Weinberg's Quantum Field Theory book I am confused about the equation (2.5.5). I'll describe the problem briefly here, but in any case, here's that page from Weinberg's book (page 64)...