Groups Definition and 867 Threads

Homework Statement Let G be a finite group and let K be normal to G. If the factor group G/K has an element of order n, show that G has an element of order n. Homework Equations None The Attempt at a Solution Lets say Kg is the element in G/K with order n. That means: (Kg)^n = K and...

say K is normal in G hence we have a factor group G/K. let g be an element of G where |g| = n. so Kg^n = K since g^n = 1. and using the properties of factor groups, we know Kg^n = (Kg)^n hence (Kg)^n = K So we know that the order of Kg divides n. Is this correct thinking? Factor groups are...

Homework Statement Hi guys, I need to explain the outlier point here, which has been shaded in the excel spread sheet when comparing the two dimensionless groups, The dimenionsless group, drag-coefficient is given by Drag/(density*V^2*D^2) and dimensionelss group, spin parameter, is given by...

Hello everyone, I am suppose to show that all the Dihedral groups (##D_n##, for ##n >2##) are noncyclic. I know that every cyclic group must be abelian. So, what I intended on showing was that at least two elements in ##D_n## are not commutative. Here are my thoughts: Because we are dealing...

I have come across physicists representing electroweak theory as some kind of decomposition (i.e. U(1)xSU(2)). I was wondering if someone could explain this 'crossing' to me a little further. Fair warning, my understanding of group/gauge theory is v rudimentary at this point.

Hello everyone, I am trying to understand the proof given in this link: https://proofwiki.org/wiki/Subgroup_of_Cyclic_Group_is_Cyclic I understand everything up until the part where they conclude that ##r## must be ##0##. Their justification for this is, that ##m## is the smallest integer...

In the theory of quantum groups Hopf algebras arise via the Fourier transform: "A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier transform" At least for nice functions, a Fourier series is just a Laurent series on a circle (which...

Hello everybody! I've just started with studying group homorphisms and tensor products, so i am still not very sure if i undertstand the subject correct. I am stuck with a question and i would ask you for some help or hints how to proceed... What i have to do is to describe...

Hi all, Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it : 1)Basically, understanding how/why the...

Homework Statement If K is normal in G and |g| = n for some g in G, show that the order of Kg in G/K divides n. Homework Equations None The Attempt at a Solution Okay so I feel like I have a solution but I don't use all the information given so I'm trying to find holes in it... g^n = 1...

Define: Euc(n) = \{ T \in End( \mathbb{R}^n )| ~ ||Tx - Ty|| = ||x - y||~\forall x,y \in \mathbb{R}^n \} This is defined as the Euclidean group of rigid motions. Can we generalize this group to be defined with any metric (well actually inner product, I suppose)? Obviously it won't be...

I need help with the proof of an apparently simple Proposition in Aigli Papantonopoulou's book: Algebra: Pure and Applied. The proposition in question is Proposition 5.2.3 and reads as follows:Can someone please help me and provide an explicit and formal proof of this proposition.Since...

Hey! :o I have to determine whether the following sentences are correct or not. Any two groups with three elements are isomorphic. At any cyclic group, each element is a generator. Each cyclic group has at least one non trivial proper subgroup. The group $G=\{ 1, i, -1, -i \}$ with respect...

I need some help with the proof of Theorem 15.18 in Fraleigh: A First Course in Abstract Algebra. The text of Theorem 15.18 reads as follows: In the above text we read: " ... ... Now $$\gamma^{-1}$$ of any non-trivial proper normal subgroup of $$G/M$$ is a proper normal subgroup of $$G$$...

A group is said to be indecomposable if it cannot be written as a product of smaller groups. An example of this is any group of prime order p, which is isomorphic to the group of integers modulo p (with addition as the group operation). Since the integers modulo p is a cyclic group (generated by...

I am revising the Isomorphism Theorems for Groups in order to better understand the Isomorphism Theorems for Modules. I need some help in understanding Dummit and Foote's proof of the Second Isomorphism Theorems for Groups (Diamond Isomorphism Theorem ? why Diamond ?). The relevant text from...

Where to go to study experimental high energy physics in the US? What are the best groups working on experiment HEP to join with?

Does anyone know of anywhere on the internet where I may find a spreadsheet containing the names of the 230 space groups? In a perfect world this would simply be a csv file containing the schoenflies and hermann-maguin notations; but I can't be too picky. Thanks

I'm currently working in a very reputable lab and am wondering if it would be wise to switch to a smaller lab if the opportunity were to arise. The reason I am considering this is because in my current lab I do mostly programming and some 3d modelling. Whereas perhaps in a smaller lab I...

I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract...

I know some groups on PDF's like CTEQ, MSTW, NNPDF... Is there any groups working on the parameterization of fragmentation functions, and publish any codes that can be called?

I want to do a survey about the parameterization of PDFs & FFs, and make a summary (say, in tables and figure) about the experiments and groups working on it, and their results. Where can I found these materials? Regards!

Alan F beardon, Algebra and Geometry chapter 1 6. For any two sets A and B the symmetric difference AΔB of A and B is the set of elements in exactly one of A and B; thus AΔB = {x ∈ A ∪ B : x / ∈ A ∩ B} = (A ∪ B)\(A ∩ B) . Let be a non-empty set and let G be the set of subsets of (note that G...

I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem. I'm learning group theory on my own, and...

Hello I've been reading some Group theory texts and would like to clarify something. Let's say we have two Lie groups A and B, with corresponding Lie algebras a and b. Does the fact that a and b share the same Lie Bracket structure, as in if we can find a map M:a->b which obeys...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with understanding Example 2.1.3 (ii) (page 39) which concerns $$L$$ as a submodule of the quotient module $$ \mathbb{Z}/p^r \mathbb{Z}$$ ... ... Example 2.1.3 (ii) (page...

Homework Statement My online notes stated that it |g| = |G| where g is an element of G then |G| is cyclic. Can somebody help me understand why this is true?

The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by ##K_{ab} = k \delta_{ab}## for some...

With groups, one often seeks to create larger groups out of smaller groups, or the reverse: break down large groups into easier-to-understand pieces. One construction often employed in this regard is the direct product. The normal way this is done is like so: The direct product of two groups...

Homework Statement Let G be a finite group with N , normal subgroup of G, and a, an element in G. Prove that if (a) intersect N = (e), then o(An) = o(a). Homework Equations The Attempt at a Solution (aN)^(o(a)) = a^(o(a)) * N = eN = N, but is the least power such that (aN)^m = N...

I'm trying to understand Čech cohomology and for this I'm looking at the example of ##S^1## defined as ##[0,1]/\sim## with ##0 \sim 1##. To compute everything, I have the cover ##\mathcal U## consisting of the sets $$U_0= (0, 2/3) \, , \, U_1= (1/3, 1) \, , \, U_2= (2/3, 1] \cup [0, 1/3)$$...

Homework Statement Find the number of ways of 9 people can be divided into two groups of 6 people and 3 people. Homework Equations The Attempt at a Solution my working is there are 6! arrangement for 6 people and 3! arrangment for 3 people. then the group of 3 people can be...

Two groups are isomorphic if they has same number of elements and if they has same number of elements of same order? Is it true? Where can I find the prove of this theorem?

I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I am currently focussed on Theorem 2.68 [page 117] concerning finitely generated groups I need help to the proof of this theorem. Theorem 2.68 and its proof read as follows:In the...

I want to get a decent introduction into group theory and Galois groups. Can somebody recommend a good book that I can use for self-study? The book of Stewart - Galois Theory looks promising.

I am reading Paolo Aluffi's book: "Algebra: Chapter 0" I am currently focussed on Chapter 2: "Groups: First Encounter". On page 41, Aluffi defines a group as follows: ------------------------------------------------------------------------------ Definition: A group is a groupoid with a...

I'm revising for my introductory particle physics exam and I've noticed SU(3) pop up a few times, it was never really explained and I don't really understand what it means. For example when talking about the conservation law of Colour, it says the symmetry is phase invariance under SU(3)...

I apologize for the informal and un-rigourous question. I have heard, in passing, that doing Galois Theory over Lie Groups instead of discrete groups is connected to solutions of differential equations instead of algebraic equations. First of all, is this correct? If so, what is this...

Hi, let S be any set and let ##Z\{S\}## be the free group on ##S##, i.e., ##Z\{S\}## is the collection of all functions of finite support on ##S##. I am trying to show that for an Abelian group ##G## , we have that : ## \mathbb Z\{S\}\otimes G \sim |S|G = \bigoplus_{ s \in S} G ##, i.e., the...

So far, I think algebraic topology is turning out to be the best thing since sliced bread. However, I'm having a bit of difficulty with homology, for one particular reason. Consider, as an example, the first homology group of ##S^1##. The definition of the free abelian group (or, in general...

Why is it that several lie groups can have the same Lie algebra? could it have to do with the space where they act transitively? Could two different Lie groups acting transitively on the same space have the same Lie algebra?

i was given that Z8 to Z4 is given by f= 0 1 2 3 4 5 6 7 0 1 2 3 0 1 2 3 where f is homomorphism. how can i find the kernal K

someone had a post on finite quotient groups. i understood that but how does one list elements of G/H if H is a subgroup of G. where: G=Z10 H={α,β,δ}

All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the...

G is a group and H is a normal subgroup of G. where G=Z6 and H=(0,3) i was told to list the elements of G/H I had: H= H+0={0,3} H+1={14} H+2={2,5} now they are saying H+3 is the same as H+0, how so?

Can one set up a multiplication table for the symmetry group C3V of the equilateral triangle. Then show that it is identical to that of the permutation group P3. I need some clarification... What about a matrix representation (2x2) for these groups? → Here was thinking to use...

I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.3 on page 37 concerns the homology groups of the Klein Bottle. Theorem 6.3 demonstrates that the homology groups for the Klein Bottle are as follows: H_1 (S) = \mathbb{Z} \oplus \mathbb{Z}/2 and H_2 (S) = 0 I...

I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.2 on page 35 concerns the homology groups of the 2-dimensional torus. Munkres shows that H_1 (T) \simeq \mathbb{Z} \oplus \mathbb{Z} and H_2 (T) \simeq \mathbb{Z} . After some work I now (just!) follow the...

If a space X is arcwise connected, then for any two points p and q in X the fundamental groups ##\pi_1(X,p)## and ##\pi_1(X,q)## are isomorphic. This means that we can, up to isomorphisms, identify both groups with their equivalence class ##\pi_1(X)##. I started to think about the generality...

Homework Statement While reading through my textbook on abstract algebra while studying for a test, I ran across the following statement: There are two isomorphism classes of groups of order 21: the class of ##C_{21}##, and the class of a group ##G## generated by two elements ##x## and...