Groups Definition and 906 Threads

Hello. I have a question that has been on my mind for some time. I always see in mathematical physics books that they identify elements of the Lie algebra with group elements "sufficiently close" to the identity. I have never seen a real good proof of this so went on an gave a proof. Let Xi be...

In M.A. Armstrongs book "Groups and Symmetry" in Chapter 12 he introduces the reader to the fascinating topic of Braid Groups. Does anyone know of a book at undergraduate level (or even a popular book) that deals with Braid Groups Can you progress with Braid Groups if you lack a...

I am reading Dummit and Foote on Representation Theory CH 18 I am struggling with the following text on page 843 - see attachment and need some help. The text I am referring to reads as follows - see attachment page 843 for details \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v +...

Homework Statement Let m and n be relatively prime positive integers. Show that if there are, up to isomorphism, r abelian groups of order m and s of order n, then there are rs abelian groups of order mn. Homework Equations The Attempt at a Solution I'm not sure how to go about...

Homework Statement Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n) Homework Equations if two groups form an isomorphism they must be onto, 1-1, and preserve the operation. The Attempt at a...

Hello everyone, I am new to this forum. Need help with this problem How many ways you can select 3-person groups from a group of 8 students? My solution: ---------------- Number of ways to make one group of 3 persons = 8C3 How do I proceed from here? Thank you.

Symmetry, Groups, Algebras, Commutators, Conserved Quantities OK, maybe this is asking too much, hopefully not. I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given. If I understand what I'm reading, there...

Homework Statement Let p be prime and let b_1 ,...,b_k be non-negative integers. Show that if : G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k} then the integers b_i are uniquely determined by G . (Hint: consider the kernel of the homomorphism f_i :G \to...

Linear Algebra Preliminaries in "Finite Reflection Groups In the Preliminaries to Grove and Benson "Finite Reflection Groups' On page 1 (see attachment) we find the following: "If \{ x_1 , x_2, ... x_n \} is a basis for V, let V_i be the subspace spanned by \{ x_1, ... , x_{i-1} ...

I am reading Grove and Benson's book on Finite Reflection Groups and am struggling with some of the basic linear algebra. Some terminology from Grove and Benson: V is a real Euclidean vector space A transformation of V is understood to be a linear transformation The group...

which is the best leaving group? Cl- enolate NR3 I couldn't figure out how to draw the enolate here, but it is a basic 3 carbon one. I believe the answer is Cl because resonance on the enolate will disperse the neg. charge. Am I on the right track or is there a factor I am missing?

Homework Statement Which of following statements are TRUE or FALSE. Why? In any group G with identity element e a) for any x in G, if x2 = e then x = e. b) for any x in G, if x2 = x then x = e. c) for any x in G there exists y in G such that x = y2. d) for any x, y in G there exists z...

I have a question where it says prove that G \cong C_3 \times C_5 when G has order 15. And I assumed that as 3 and 5 are co-prime then C_{15} \cong C_3 \times C_5 , which would mean that G \cong C_{15} ? So every group of order 15 is isomorohic to a cyclic group of order 15...

I'm looking for an algorithm for dividing a set of numbers into groups, and then doing it again in such a way that no numbers are in a group together more than once. For instance if you have 18 numbers and divide them into groups of three you should be able to do this 8 times without any...

Homework Statement The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so...

The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so I am pretty lost in...

Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties? So for example for a group G with order 15 to show that G \cong C_3 \times C_5 would I just have to define all the possible transformations to define the isomorphism...

Homework Statement Let G be a group with identity e, and suppose that a, b, c, x in G. Determine x, given that x2a=bxc-1 and acx = xac. Homework Equations The Attempt at a Solution I know the three axioms for group. G1. Associativity. For all a, b, c in G, (a * b) * c = a...

If we have two groups of photons; each group consisting of two entangled photons. We allow one of the photons from each group to interact with another object. If we perform a polarisation measurement on each photon of one group, will the photons in the other group, independent of whether...

Homework Statement We are supposed to say how many limit points the set A={sin(n)} where n is a positive integer. My teacher said to use a theorem by Kronecker to help with it. His theorem says from wiki, that an infinite cyclic subgroup of the unit circle group is a dense subset...

I am reading Dummit and Foote Ch 18, trying to understand the basics of Representation Theory. I need help with clarifying Example 3 on page 844 in the particular case of S_3 . (see the attahment and see page 844 - example 3) Giving the case for S_3 in the example we have the...

The problem is kind of easy to understand. Given is some points, say 10 points. (I am using numbering for understanding) 0 1 2 3 4 5 6 7 8 9 Now group these such that the group size is 5 and there is no overlap so, there can be 2 groups. the groups are (0 1 2 3 4) & (5 6 7 8 9) Now...

Help! For p prime I need to show that C_{p^2} \ncong C_p \times C_p where C_p is the cyclic group of order p. But I've realized I don't actually understand how a group with single elements can be isomorphic to a group with ordered pairs! Any hints to get me started?

Regarding finite cyclic groups, if a group G, has generator g, then every element h \in G can be written as h = g^k for some k. But surely every element in G is a generator as for any k , (g^k)^n eventually equals all the elements of G as n in takes each integer in turn. Thanks...

HI, I was reading an article and it says that a finite group of order p^aq^b, where p, q are primes, is solvable and therefore not simple. But I can't quite understand why this is so. I do recall a theorem called Burnside's theorem which says that a group of such order is solvable. But then I...

Homework Statement Let G be a cyclic group of order n and let k be an integer relatively prime to n. Prove that the map x\mapsto x^k is sujective. Homework Equations The Attempt at a Solution I am trying to prove the contrapositon but I am not sure about one thing: If the map is...

Apparently there is an isomorphism between the additive group (ℝ,+) of real numbers and the multiplicative group (ℝ_{>0},×) of positive real numbers. But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?! Thanks for...

Hi, I have a mathematics/Matlab question. Suppose I have a speaker that serves as a sound source, and two IDENTICAL microphones to the left and right of this speaker. Suppose that each microphone collects data regarding the sound level of the speaker, and that there are over 3,000 data values...

Homework Statement Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) } and N=<(13)(24)> which is a normal subgroup of d4 . List the elements of d4/N . Homework Equations The Attempt at a Solution I computed the left and right cosets to...

how does one evaluate the direct product between a group G with components that are say 2-tuple and a group H with components that are just 1-tuple?

If I were to solve a system of multiple equations in the form αx+βy+ζz=p_{1} Where α,β,ζ are constants x,y,z are variables, and p is a prime, how would I use Galois theory and/or number theory to find the number of solutions if the other equations could all be written in the form...

Hello everyone, When we speak about the SU(2)L group (in electroweak interactions for example), about what group do we talk ? What is the difference with the SU(2) group ? And with the SU(2)R ? Why is the label so important ? I ask this because I see that a Lagrangien can be invariant...

Let G and H be finite groups. The maximal subgroups of GxH are of the form GxM where M is maximal subgroup of H or NxG where N is a maximal subgroup of G. Is this true?

I was thinking about some similarities in the definitions of group and field, and if it would be possible to generalize in some sense, like follows. A field is basically a set F, such that (F,+) is a commutative group with identity 0, and (F-{0}, .) is a commutative group with identity 1, and...

Hello I have to show that the free product of a collection of more than one non-trivial group is non-abelian. But doesn't this just follow from the definition of the free product? Or how would you tackle this question?

Homework Statement Let G be a group with identity e. Let a and b be elements of G with a≠e, b≠e, (a^5)=e, and (aba^-1)=b^2. If b≠e, find the order of b. Homework Equations Maybe the statement if |a|=n and (a^m)=e, then n|m. Other ways of writing (aba^-1)=b^2: ab=(b^2)a...

I am in the fourth year of an MPhys and feel a PhD is the best way to further myself. I want to apply to groups that specialize in the theory and implementation of computational modelling. I would prefer this to be a group with a wide range across several branches of Physics as opposed to a...

The question asks: 3) Let X be the set of 2-dimensional subspaces of F_{p}^{n}, where n >= 2. (a) Compute the order of X. (b) Compute the stabilizer S in GL_{n}(F_{p}) of the 2-dimensional subspace U = {(x1, x2, 0, . . . , 0) ε F_{p}^{n} | x1, x2 ε F_{p}}. (3) Compute the order of S. (4)...

WTS is that \mathbb R^*/N \ \cong \ \mathbb R^{**} where N = (-1, 1) then prove that \mathbb R^*/\mathbb R^{**} \ is \ \cong \ to \mathbb Z/2\mathbb Z So the best answer in my opinion is to construct a surjection and use the first iso thm. f:\mathbb R^*\rightarrow\mathbb R^{**}...

what does it mean. I'm thinking list all groups of such order for instance. 115 = 5* 23 hence Z5⊕ Z23 ≈ Z115 ?

If |Aut H| = 1 then how can I show H is Abelian? I've shown a mapping is an element of Aut H previously but didn't think that would help. I have been looking through properties and theorems linked to Abelian groups but so far have had no luck finding anything that would help. The closest I...

Are all groups nonempty? If so, is it because all groups have an identity (element)?

I'm wondering if anyone can help me with learning how to write groups as an external direct product of cyclic groups. The example I'm looking at is for the subset {1, -1, i, -i} of complex numbers which is a group under complex multiplication. How do I express it as an external direct...

Homework Statement G is isomorphic to H. Prove that if G has a subgroup of order n, H has a subgroup of order n. Homework Equations G is isomorphic to H means there is an operation preserving bijection from G to H. The Attempt at a Solution I don't know if this is the right...

Homework Statement Show that the product of two infinite cyclic groups is not an infinite cyclic? Homework Equations Prop 2.11.4: Let H and K be subgroups of a group G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. then f is an isomorphism iff H intersect K is...

So this isn't really a homework question and I'm sorry if I posted in the wrong section. Please feel free to move it if needed. I'm doing a project on groups. I will be working on groups of order 12 through 16. I have no questions to follow or really have no idea on what to do with them...

I am currently looking at what research groups to apply to for starting a PhD next year, is there anywhere particularly good that I am missing out? I did my Masters in Theoretical Physics but I am actually leaning more towards a mixture of both experimental and theory at this stage, I did do...

This is not a homework question, just a question that popped into my head over the weekend. My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set...

Factor groups! Please I just want to ask about factor groups.. how could a factor group G/A acts on A by conjugation, knowing that A is a normal & abelian subgroup of G.. and what do we mean when we say that an element in a group has jus 2 conjugates?? thanks in advance :)

(This is my first post on PF btw - I posted on this another thread, but I'm not sure if I was supposed to) I was doing some practice problems for my exam next week and I could not figure this out. Homework Statement Suppose a is a group element such that |a^28| = 10 and |a^22| = 20...