Groups Definition and 867 Threads

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition of...

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition...

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of Definition 1.4, Page 12 ... ... The start of Section 1.5, including Definition 1.4 ... reads as...

I’ve read about the exponential map that for Lie groups the exponential map is actually the exponential function. But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric. But usually nobody gives you a metric with a Lie group. So how can I get...

Homework Statement Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G. I am confused as to why the proof makes the assumption that <a> is a subset of the set G. Homework EquationsThe Attempt at a Solution The proof I think is like the...

So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following: Consider the Klein four group, if we...

So it's said that every group is a symmetry group of some tangible object. For example, ##S_3## is the symmetry group of ##\{1,2,3 \}##, and ##D_{2n}## is the symmetry group of an n-gon. But what is ##GL_{10} (\mathbb{R})## the symmetry group of? What about ##\mathbb{Z}##? I have found two...

Homework Statement Prove that any two cyclic groups of the same finite order are isomorphic Homework EquationsThe Attempt at a Solution So I began by looking at the map ##\phi : \langle x \rangle \to \langle y \rangle##, where ##\phi (x^k) = y^k##. So, I went through and showed that this is...

I am trying to show that ##\mathbb{R} - \{ 0\}## is not isomorphic to ##\mathbb{C} - \{0 \}##. If we simply look at ##x^3 = 1##, it's clear that ##\mathbb{R} - \{ 0\}## has one solution while ##\mathbb{C} - \{0 \}## has three. My question, how can I use ##x^2 = -1## to show that they are not...

Given a group, can we regard its elements as statements of what the operation does, while its representations are the mathematical translation? For instance, given a square, I'd say that ##a## is an operation that rotates the square by 90°, and the representation of ##a## would be the matrix...

This article describes how total carbon in organisms is distributed among different groups. Science news article here. Original PNAS article here. Plants win, not surprisingly (primary producers). Bacteria are next (Archaea which are similar to bacteria in many ways have much less carbon)...

How many isomers are there with the following description? - Thioesters with the formula ${C}_{4}{H}_{8}OS$? I was able to draw 2 of them, but apparently, the answer key showed and stated that there are 4. I am confused about why the following two are possibliities: I thought that thioesters...

Hi all I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of...

I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows...

I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows: In the above proof by...

I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows: In the above proof by...

I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows: In the above proof by...

In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...

I'm reading "Division Algebras and Quantum Theory" by John Baez https://arxiv.org/abs/1101.5690 In the last paragraph of section 5 (Applications) he says the following "SU(2) is not the only compact Lie group with the property that all its irreducible continuous unitary representations on...

Who really wrote the best introductory account of representation theory in QM that I've seen so far ? [Likely mis-attribution discussed here below; prefixed "Advanced" to reach lecturers who are more likely to know the answer to this question.] It's available via...

The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group. What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term? I've been trying to find literature on the groups for the...

Homework Statement For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms. Homework Equations (R,◦), where x◦y=2xy+1 (R*,◦), where x◦y=πxy and R* = R - {0} The Attempt at a Solution For question 1, I found a G2...

We have the 3 equivalent definition for solvable groups: There exists a chain of subgroups 1 < G1 ...< Gi + < G i+1 < Gr = G such that Gi is normal in Gi+1 and Gi+1/Gi is abelian. Another definition is there exists 1 < H1 ...< Hi + < H i+1 < Hs = H such that Hi is normal in Hi+1, and...

I would kindly appreciate any corrections to my conclusions, because I need to get this subject straight for learning QFT in a satisfactory way. From what I have been reading about Lie groups so far, I have concluded the following: 1 - A group is independent of a representation, but we usually...

Homework Statement [/B] ##X## is linearly independent if and only if every nonzero element of the subgroup ##\langle X \rangle## may be written uniquely in the form ##n_1 x_1 + ... n_k x_k## (##n_i \in \Bbb{Z} \setminus \{0\}##, and ##x_1,...,x_k \in X## are distinct). Homework Equations [/B]...

How can you prove that there can only be 2 possible four-element group?

hi gang there are currently 2 significant spot groups visible traversing the face of the solar disk The centre-left string is active region 2671 and the region near the right edge ( eastern limb) is active region 2672. AR2672 will continue to rotate across the disk across the next 2 weeks...

I would like to find the 12 to 19 abortion rate per 10,000 women from the following table: This would be the merging of age groups "12-17" with "18-19" to some how get the 12-19 abortion rate. EDIT: "Distribution of abortion age" was translated from "Répartition des ges l'avortement" in...

Hi, I am looking for textbooks in QFT. I studied QFT using Peskin And Schroeder + two year master's degree QFT programme. I want to know about the next items: 1) Lorentz group and Lie group (precise adjectives, group representation and connection between fields and spins from the standpoint of...

I'm not sure if this question belongs to here, but here it goes Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the Example 47.7 ... Example 47.7 and its proof read as follows: In the above example, Anderson and Feil write the following...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the Example 47.7 ... Example 47.7 and its proof read as follows: In the above example, Anderson and Feil write the following: "...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the proof of Theorem 47.1 ... Theorem 47.1 and its proof read as follows: At the end of the above proof by Anderson and Feil, we...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the proof of Theorem 47.1 ... Theorem 47.1 and its proof read as follows: At the end of the above proof by Anderson and Feil, we...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that $$\text{Aut} ( \mathbb{C} ) $$is a group with...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that ##\text{Aut} ( \mathbb{C} )## is a group with only...

Homework Statement Let G be a group. if H = ##{x \epsilon G : x = x^{-1}}##, that is H consists of all elements of G which are their own inverses, prove that H is a subgroup of G. Homework Equations to show H is a subgroup of G, show that H is closed under the operation of G and every element...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that $$\text{Aut}( \mathbb{R} )$$ is the trivial...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that ##\text{Aut}( \mathbb{R} )## is the trivial group...

Ok, so in a logistic regression context, I need to test if the probability of ##Y_{i}=1 ## is the same for two different groups at different ages where age is a continuous variable. This is actually complicated because of nonlinearity. Can I default to testing if the odds of ##Y_{i}=1 ## is...

Homework Statement I am trying to show that neither ##Z_{p^n}## nor ##\mathbb{Z}## can be written as any family of its proper subgroups. Homework EquationsThe Attempt at a Solution First, I believe this solution (http://www.auburn.edu/~huanghu/math7310/7310-hw2-answer.pdf see problem 6) is...

Homework Statement Are the groups ##\mathbb{Z}_8 \times \mathbb{Z}_{10} \times \mathbb{Z}_{24}## and ##\mathbb{Z}_4 \times \mathbb{Z}_{12} \times \mathbb{Z}_{40}## isomorphic? Why or why not? Homework EquationsThe Attempt at a Solution I think I am misunderstanding the Theorem of Finitely...

Say for example I want to see the rate of injury for firefighter vs police vs soldier. ##InjuryCount_{i}## The number of injuries recorded for the ith person over time ##T_{i} ## Time the person was followed. Varies from person to person. ##I(f)_{i}## indicator for ith person of being a...

Homework Statement Let ##A## and ##B## be finite groups, and ##A \times B## be their direct product. Given that ##(a,1)## and ##(1,b)## commute, and that ##(a,1)^n = (a^n,1)## and ##(1,b)^n = (1,b^n)## for all a and b, show that the order of ##(a,b)## is the least common multiple of the orders...

I am looking for some general guidance on questions of the form: "Using a ## \Delta ## complex, compute the homology groups of the quotient space obtained fromt the 2-sphere ##S^2## by identifying three of its distinct points." Similarly I have a question about a torus with two points...

Enjoy! http://www.math.columbia.edu/~woit/QM/qmbook.pdf

hi Show that Z2 X Z3 IS ISOMORPHIC TO THE GROUP G = (1,2,4,8,10)

hello every one . can someone please find the left invariant vector fields or the generator of SO(2) using Dr. Frederic P. Schuller method ( push-forward,composition of maps and other stuff) Dr Frederic found the left invariant vector fields of SL(2,C) and then translated them to the identity...

https://en.wikipedia.org/wiki/List_of_English_terms_of_venery,_by_animal Lists lots of animals. Clearly people made up some of these things when it was clear nobody really knew if it was correct or not. Heck, we can do that! So let's see what we can come of with: How about a slime of...

Homework Statement Consider G = {1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64} with the operation being multiplication mod 65. By the classification of finite abelian groups, this is isomorphic to a direct product of cyclic groups. Which direct product? Homework EquationsThe...