Groups Definition and 906 Threads

Homework Statement Prove that isomorphism is an equivalence relation on groups. Homework Equations Need to prove reflexivity, symmetry, and transitivity for equivalence relationship to be upheld. **We will use ≅ to define isomorphic to** The Attempt at a Solution Let G, H, and K be groups...

Homework Statement e) If H ∼= Z3 × Z3 show that there are exactly 2 conjugacy classes of elements of order 2 in Aut(Z3 × Z3) = GL(2, Z3). f) Choosing an element of each conjugacy class in e), construct two semidirect products of H and K. By counting orders of elements in each such group, show...

Hello, Given a Lie group G and a smooth path γ:[-ε,ε]→G centered at g∈G (i.e., γ(0)=g), and assuming I have a chart Φ:G→U⊂ℝn, how do I define the derivative \frac{d\gamma}{dt}\mid_{t=0} ? I already know that many books define the derivative of matrix Lie groups in terms of an "infinitesimal...

Hello, I'm reading a book on Lie group theory, and before giving the definition of a Lie group G, the author defines the concept of chart as a pair (U(g), f) where: i) U(g) is a neighborhood of g∈G ii) f : U(g)→f(U(g))⊂ℝn is an invertible map such that f(U(g)) is an open subset of ℝn. My...

1. The problem statement, all variables and given/known. Rank the following leaving group in order of increasing ability to leave? A) H2O B) NH2 C) OH D) I E) NH3 Homework Equations Also not entirely sure the order of H2O and NH3. What should I be looking for to answer this portion of the...

Hello, I am having some trouble truly interpreting what certain notation means when defining quotient groups, etc. (My deepest apologies in advance, with my college workload I simply have not had the time to really sit down and master latex.) Here are a few random examples I've seen in...

Hi, the abelianization of a group G is given by the quotient G/[G,G], where [G,G] is the commutator subgroup of G. When dealing with finite groups, the commutator subgroup is given by the (normal) subgroup generated by all the commutators of G. If we consider instead the case of G being a Lie...

Hello everybody, As I mentioned in the title, it is about molecular symmetry and its Hamiltonian. My question is simple: For any molecule that belong to a precise point symmetry group. Is the Hamiltonian of this molecule commute with all the symmetry element of its point symmetry group...

Hello, I was given a question (not a HW question..) in which i was asked to calculate the number of ways to sort n numbers into k groups, where for any two groups, the elements of one group are all smaller or larger than the elements of the other group. The answer is supposed to be...

9 greeks, 17 finns, 7 russians, 11 chinese and 8 swedish students are studying in groups. A group can consist of one or more persons. If a group has two or more persons of the same nationality, it must also have at least one person representing another nationality. The question is: in how many...

It all starts with a shuffling. We have a sorted list, we take out groups of numbers and we insert them in random positions: Then, it is about sorting the numbers back using the same approach, in a minimum numbers of steps (which doesn't have to be identical to a number of steps taken during...

How many ways can you arrange 52 things into 4 groups BUT the groups do not have to be the same size?!?

Hey, guys! I was recently reading (attempting) about spin groups. I heard a little bit about SO(3), but still don't know much. I was wondering if someone could explain what a spin group is and why it is useful? Is there some way to visualize spin groups? Please note: I know literally nothing...

Homework Statement How many non-isomorphic groups of two elements are there? Homework EquationsThe Attempt at a Solution I don't understand exactly what we are being asked. If we have a group of two elements under, say, addition, then G =\{0, g\}. Then also g+g = 0 must be true, means g is its...

Is there a physical reason why all gauge groups considered in SM and especially beyond are always semisimple? [+ U(1)] What would happen if they were solvable?

How to draw the symettry axis for 1-fold rotation ? And why C1v is identical to C1h ? Thanks

I have made two posts recently concerning the composition series of groups and have received considerable help from Euge and Deveno regarding this topic ... in particular, Euge and Deveno have pointed out the role of the Correspondence Theorem for Groups (Lattice Isomorphism Theorem for Groups)...

Homework Statement Consider the ring of polynomails in two variables over a field K: R=K[x,y] a)Show the elements x and y are relatively prime b) Show that it is not possible to write 1=p(x,y)x+q(x,y)y with p,q \in R c) Show R is not a principle ideal domain Homework Equations None The...

I am reading Paolo Aluffi's book, Algebra: Chapter 0 ... I am currently focused on Chapter 4, Section 3: Composition Series and Solvability ... I need help with an aspect of Aluffi's proof of the Jordan-Holder Theorem (Theorem 3.2, page 206) which reads as follows: Theorem 3.2 and the early...

I am reading Paolo Aluffi's book, Algebra: Chapter 0 ... I am currently focused on Chapter 4, Section 3: Composition Series and Solvability ... I need help with Exercise 3.3 on page 213, which reads as follows: I hope someone can help ... and in so doing use Aluffi's notation ... So that MHB...

Homework Statement Good day, I need to show: \mathbb{Z}_{4}\oplus \mathbb{Z}_{4} is not isomorphic to \mathbb{Z}_{4}\oplus \mathbb{Z}_{2}\oplus \mathbb{Z}_{2} Homework Equations None The Attempt at a Solution I was given the hint that to look at the elements of order 4 in a group. I know...

I know of only one group, ##A_4## of order 12 which does not have a subgroup with order dividing the group size. In this case, a subgroup of size ##6##. What property of a group causes this? Would I expect to find other examples only in non-abelian groups or are there abelian groups which do...

Hi, I was wondering how to find a minimal set of generators for the symmetric groups. Would it be difficult to fill-in the following table? ##\begin{array}{cl} S_3&=\big<(1\;2),(2\;3)\big> \\ S_4&=\big<(1\;2\;3\;4),(1\;2\;4\;3)\big>\\ \vdots\\ S_{500} \end{array} ## Is there a procedure to...

Hi, I was trying to identify some infinite non-abelian groups other than ##GL_n(G)## and also other than contrived groups such as the group: ##G=\big<r,s : r^2=s^3=1\big>## as per...

Hi, I am learning classical mechanics right now, Particularly Noether's theorem. What I understood was that those kinds of transformations under which the the Hamiltonian framework remains unchanged, were the key to finding constants of motion. But here are my Questions: 1. What is...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently revising Section 2.6 Quotient Groups in order to understand rings better ...I have another question regarding the proof of Proposition 2.123 part (i) ... which I think...

I have just received some help from Euge regarding the proof of part of the Correspondence Theorem (Lattice Isomorphism Theorem) for groups ... But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ... We are to consider...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently revising Section 2.6 Quotient Groups in order to understand rings better ... I need help with understanding the proof of Proposition 2.123 part (i) ... which I think is...

If there a finite field where both group structures have hard discrete logs? Discrete log in the additive group means multiplicative inverse.

If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...

Hi I am a physics graduate student. Recently I am learning representation theory of groups. I understand the basic concepts. But I need a good book with lots of examples in it and also exercise problems on representation theory so that I can brush up my knowledge.The text we follow is "Lie...

Homework Statement so for a side task I'm supposed to assign people to groups for an icebreaker in python, can anyone give me links to theories that I could read up on or give me suggestion X number of people at my company signed up for a dinner roulette as a way to meet new people. Everyone...

Homework Statement Let G = G1 × G2 be the direct product of two simple groups. Prove that every normal subgroup of G is isomorphic to G, G1, G2, or the trivial subgroup. The Attempt at a Solution I tried proving that the normal subgroups would have to be of the form Normal subgroup X Normal...

I am having problem working with the objects in the title. Working with permutations, rotations and reflections is fine, but I have problem with the following: Showing a subgroup is or is not normal (usually worse in the case of symmetric groups) Finding a subgroup of order n. Showing that...

I feel there should be a way to find non-trivial (other than ##\operatorname{inn} G##) automorphisms of these groups other than by trial-and-error computation. Take for example ##\operatorname{aut}\mathbb{Z}_{100!}^*## . Are these just computationally inaccessable to us forever? How about a...

I am attempting to answer the attached question. I have completed parts 1-4 and am struggling with part 5. 5. Prove that if a^{l_0}b_1^{l_1}...b_n^{l_n}=e then a^{l_0}=b_1^{l_1}=...=b_n^{l_n}=e If |a|>|b1|>|b2|>...>|bn| then I could raise both sides of a^{l_0}b_1^{l_1}...b_n^{l_n}=e to the...

Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...

Let ##G## be a group and let ##R## be the set of reals. Consider the set ## R(G) = \{ f : G \rightarrow R \, | f(a) \neq 0 ## for finitely many ## a \in G \} ##. For ## f, g \in R(G) ##, define ## (f+g)(a) = f(a) + g(a) ## and ## (f * g)(a) = \sum_{b \in G} f(b)g(b^{-1}a) ##. Prove that ##...

Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?

If σ is a k-cycle with k odd, prove that there is a cycle τ such that τ^2=σ. I know that every cycle in Sn is the product of disjoint cycles as well as the product of transpositions; however, I'm not sure if using these facts would help me with this proof. Could anyone point me in the right...

Homework Statement Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...

∴Homework Statement Let ℝ>0 together with multiplication denote the reals greater than zero, be an abelian group. let (R>0)^n denote the n-fold Cartesian product of R>0 with itself. furthermore, let a ∈ Q and b ∈ (ℝ>0)^n we put a⊗b = (b_1)^a + (b_2)^a + ... + (b_n)^a show that the abelian...

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 2 in Chapter 3, namely: "2. Several Characterizations of the Orthogonal Groups". I need help in fully understanding the proof of Proposition 3.10. Section 2 in Ch. 3...

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 2 in Chapter 3, namely: "2. Several Characterizations of the Orthogonal Groups". I need help in fully understanding some important remarks following Proposition 3.10...

Homework Statement The set ℝ^2 with vector addiction forms an abelian group. a ∈ ℝ, x = \binom{p}{q} we put: a ⊗ x = \binom{ap}{0} ∈ ℝ^2; this defines scalar multiplication ℝ × ℝ^2 → ℝ^2 (p, x) → (p ⊗ x) of the field ℝ on ℝ^2. Determine which of the axioms defining a...

I am looking for a good text at senior undergraduate level on Lie Theory, and in particular, Lie Groups ... Does anyone have any suggestions? Peter

Hello, some weeks ago I was having a first look at the world of crystals: http://en.wikipedia.org/wiki/Crystal_system Now I forgot the bit that I've understood but before trying to study the topic again I would like to ask an other simple question: " What makes the 32 crystallographic point...

Homework Statement "The Daily Planet ran a recent story about Kryptonite poisoning in the water supply after a recent event in Metropolis. Their usual field reporter, Clark Kent, called in sick and so Lois Lane reported the stories. Researchers plan to sample 288 individuals from Metropolis...

Hi, I am trying to find all groups G of order 16 so that for every y in G, we have y+y+y+y=0. My thought is using the structure theorem for finitely-generated PIDs. So I can find 3: ## \mathbb Z_4 \times \mathbb Z_4##, ## \mathbb Z_4 \times \mathbb Z_2 \times \mathbb Z_2 ## , and: ##...

What is the smallest positive integer n such that there are exactly 3 nonisomorphic Abelian group of order n