Groups Definition and 906 Threads

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...

Homework Statement In lab we synthesised cis and trans copper glycine and we have to use IR to differentiate the two so we have to figure out the number of IR active vibrations for each complex. It's been a year since I did anything with point groups so I'm not sure if I did it right. Homework...

From my understanding of semiconductors, we are able to create semiconductors by combining different group of elements that fulfill the octet rule to produce covalent bonds and where their electronegativities provide a energy band gap that is between that of a conductor and insulator. Why is the...

We define a cyclic group to be one all of whose elements can be written as "powers" of a single element, so G is cyclic if ##G= \{a^n ~|~ n \in \mathbb{Z} \}## for some ##a \in G##. Is it true that in this case, ##G = \{ a^0, a^1, a^2, ... , a^{n-1} \}##? If so, why? And why do we write a cyclic...

To show that a subset of a group is a subgroup, we show that there is the identity element, that the subset is closed under the induced binary operation, and that each element of the subset has an inverse in the subset. My question is regarding showing closure. To show that the subset is closed...

Homework Statement Show that ##\langle \mathbb{R}_{2 \pi}, +_{2 \pi} \rangle## is not isomorphic to ##\langle \mathbb{R}, +\rangle## Homework EquationsThe Attempt at a Solution I know how to show that two groups are isomorphic: by finding an isomorphism between them. However, I am not sure how...

Homework Statement Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). Homework EquationsThe Attempt at a Solution So I know the properties that define a group are...

The principal connection contrary to other connections like the affine connection has a tensorial character respect to the principal bundle, does thin mean that if the principal connection is not trivial it follows that the principal bundle isn't trivial either(unlike the case with affine...

Homework Statement Let G be an abelian group of order n, and let k be an nonnegative integer. If k is relatively prime to n, show that the subgroup generated by a is equal to the subgroup generated by ak Homework EquationsThe Attempt at a Solution I'm not sure where to start. I know that we...

Homework Statement Show that ##\mathbb{Z}_8^*## and ##\mathbb{Z}_12^*## are isomorphic, where ##\mathbb{Z}_n^* = \{x \in \mathbb{Z} ~|~ \exists a \in \mathbb{Z}_n(ax \equiv 1~(mod~n)) \}##, and the group operation is regular multiplication. Homework EquationsThe Attempt at a Solution We can...

I've run across a Lie group notation that I am unfamiliar with and having trouble googling (since google won't seem to search on * characters literally). Does anyone know the definition of the "star groups" notated e.g. SU*(N), SO*(N) ?? The paper I am reading states for example that SO(5,1)...

I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy) I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix. (I understand how I obtain this equation... that is not the issue.) Now I am making the leap to learning about...

Homework Statement Show that if ##G = \langle x \rangle## is a cyclic group of order ##n \ge 1##, then a subgroup ##H## is maximal; if and only if ##H = \langle x^p \rangle## for some prime ##p## dividing ##n## Homework Equations A subgroup ##H## is called maximal if ##H \neq G## and the only...

I've been looking at John Baez's lecture notes "Lie Theory Through Examples". In the first chapter, he says Dynkin diagrams classify various types of object, including "simply-connected, complex, simple Lie groups." He discusses the An case in detail. But what are the simply-connected, complex...

Hi! I'm studying Lie Algebras and Lie Groups. I'm using Brian Hall's book, which focuses on matrix lie groups for a start, and I'm loving it. However, I'm really having a hard time connecting what he does with what physicists do (which I never really understood)... Here goes one of my questions...

Thought experiment: Assume two galaxies in a galaxy group, initially at rest (with respect to one another). The distance between the centers of the galaxies is r = 1 Mpc. The total mass of each galaxy is mg = 6 ∙ 1042 kg (including dark matter). This is ≈ 3 ∙ 1012 solar masses. The...

This is an old qual question, and I want to see if I have it right. I had virtually no instruction in homology despite this being about 1/4 of our qualifying exam, so I am feeling a bit stupid and frustrated. Anyway, I am given a space defined by three polygons with directed edges as...

This question was originally posted by ElConquistador, but in my haste I mistakenly deleted it as a duplicate. My apologies... For part (a) we can define two cyclic subgroups of order $2$, both normal, $\langle J\rangle$ and $\langle K\rangle$ such that $V=\langle J\rangle \langle K\rangle$...

Suppose $G$ is an infinite group and $H$ is an infinite subgroup of $G$. Let $g\in G$. Suppose $\forall h\in H\ \exists h'\in H$ such that $gh=h'g$. Can we conclude that $gH=Hg$? What if $G$ and $H$ are of finite orders?

I am trying to learn about the various SU groups related to QCD. I have about 5 QFT and Particle physics books from my student library and written down about 20 pages of handwritten notes about specific parts of say generators, matrices, group properties etc. - but i don't really feel that I...

Homework Statement Let c be a pth root of unit where p is prime. Then the Galois group G(Q(c):Q) is isomorphic to Z_p*. Show that if there is some m that divides p-1, then there is an extension K of Q such that G(K:Q) is isomorphic to Z_q* Homework EquationsThe Attempt at a Solution I suspect...

Hi, I have problems understanding why, for example, a Yd1 connection introduces a -30 degree phase shift, see image below. How should I think when I want to produce vector groups like that, and derive myself what the phase shift should be? Thanks..

Homework Statement How many groups of 5 dances couples can be formed from a pool of {12M, 10F}? Homework Equations {}^n\!P_k = \frac{n!}{(n-k)!} \\ {}^n\!C_k = \frac{n!}{k!(n-k)!} The Attempt at a Solution We were shown one solution in class which is to find the number of groups of 5M that...

Homework Statement For chemistry I have to name alkanes, however there is this one that I am unsure of. This one consists of both alkyl groups and halogens. Would I make my alkyl group the lowest possible number or my halogen group, or does it not matter as long as I use the lowest possible...

Hi. I am looking for a QM book that covers symmetry , time-reversal , angular momentum representations in SO(3). I have a few books and most of them don't have much detail on these subjects.The main one that does is Sakurai. Any other suggestions ? Thanks

Homework Statement Let n>=2 n is natural and set x=(1,2,3,...,n) and y=(1,2). Show that Sym(n)=<x,y> Homework EquationsThe Attempt at a Solution Approach: Induction Proof: Base case n=2 x=(1,2) y=(1,2) Sym(2)={Id,(1,2)} (1,2)=x and Id=xy so base case holds Inductive step assume...

I am seriously considering switching research groups. To be brief: I am a graduate student in condensed matter physics who had just passed the PHD candidate qualification exam. I have been working in my current lab for a year. I am unsatisfied with my working conditions in terms of interactions...

So I've been thinking of continuing after my MSc degree to do a PhD. But I have trouble getting a good feel on research groups of quantum optics and nanophotonics in the world. So my situation is: I life in the Netherlands and I have a good feel on most research groups in the country, but I'm...

Suppose one was working for a Professor. Then you mutually decide that it's not a good fit. The professor has been helpful in terms of suggestions for other people to work with and you found another professor who has given you some reading material earlier. How do I email the new professor if...

Homework Statement Homework EquationsThe Attempt at a Solution I counted 4 functional groups. I got: -Carboxylic acid -Ketone -Alcohol -Ether However, this combination is not available. I was wondering if phenol is a functional group as C seems the most likely option. I thought phenyl is a...

micromass submitted a new PF Insights post Groups and Geometry Continue reading the Original PF Insights Post.

Hi, I trying to understand. If there is non-trivial SU(3) group, is it always possible to find SU(2) as part of SU(3)? And same question about SU(2) and U(1).

Hello! I watched a video on the Youtube channel Kurzgesagt titled How far can we go? Limits of humanity The video attempts to explain why we may be limited to our local galaxy group even with science fiction technologies. During a part of the video (starting at 2:26), they try to explain how...

I am reading Donald S. Passmore's book "A Course in Ring Theory" ... I am currently focussed on Chapter 9 Tensor Products ... ... I need help in order to get a full understanding of the free abelian group involved in the construction of the tensor product ... ... The text by Passmore...

Hey! :o Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find for the ideal $I=2\mathbb{Z}$ of $\mathbb{Z}$ the $\{m\in M\mid am=0, \forall a\in I\}$ as a product of cyclic groups. We have the following...

I am reading Paolo Aluffi's book: Algebra: Chapter 0 ... ... I am currently focussed on Section 5.4 Free Abelian Groups ... ... I need help with an aspect of Aluffi's preamble to introduce Proposition 5.6 ... Proposition 5.6 and its preamble reads as follows: In the above text from Aluffi's...

I am reading Paolo Aluffi's book: Algebra: Chapter 0 ... ... I am currently focussed on Section 5.4 Free Abelian Groups ... ... I need help with an aspect of Aluffi's preamble to introduce Proposition 5.6 ... Proposition 5.6 and its preamble reads as follows: In the above text from Aluffi's...

Often, in the study of algebraic objects certain things (like tensor products) are often defined primarily in terms of an universal mapping property. When one is used to "concrete objects" one can calculate with, this often comes as a shock to the system. One feels as if one is spinning...

I'm new to proofs and I'm not sure from which assumptions one has to start with in a proof. I'm trying to prove the generalized associative law for groups and if I start with the axioms of a group as the assumptions then I already have the proof. From what basic assumptions should one start...

Homework Statement Write ##C_3\langle x|x^3=1\rangle## and ##C_2=\langle y|y^2=1\rangle## Let ##h_1,h_2:C_2\rightarrow \text{ Aut}(C_3\times C_3)## be the following homomorphisms: $$h_1(y)(x^a,x^b)=(x^{-a},x^{-b})~;~~~~~~h_2(y)(x^a,x^b)=(x^b,x^a)$$ Put ##G(1)=(C_3\times C_3)\rtimes_{h_1}C_2...

Hi, For a SoC project I am working on I need to select one cell which is most critical. example, If a bus is going through 1000 stops 1000 times I have mean median mode of delay contribution of that stop compared to the total delay to reach from start to stop point and N (number of times bus...

Question This is what I have done so far. I was wondering if anyone could verify that I found the correct minimum polynomial and roots? If I am incorrect, could someone please help me by explaining how I would find the min polynomial and roots? Thank you.

Let f(x)=x4-2x2+9 Find the splitting field and Galois group for f(x) over ℚ Here is what I have written out so far. If I have found the splitting field E correctly, have I proceeded with the Gal(E/F) group correctly? Also, how would I go about finding the roots of this equation by hand...

I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 6.3 A Word on Free Groups ... I have a basic question regarding the nature and character of free groups ... Dummit and Foote's introduction to free groups reads as follows: In the above text...

I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 6.3 A Word on Free Groups ... I have a basic question regarding the nature and character of free groups ... Dummit and Foote's introduction to free groups reads as...

Hey! :o Let $G$ be a finite group with the following property: for each two of its subgroups $X,Y\subseteq G$ it holds either $X\cap Y=1$ or $X\subseteq Y$ or $Y\subseteq X$. I want to show the following: If $H\leq G$ then either $|H|$ is a power of a prime or $|H|$ and $|G:H|$ are...

Hey! :o I want to show that if $G$ is of order $2p$ with $p$ a prime, then $G\cong \mathbb{Z}_{2p}$ or $G\cong D_p$. I have done the following: We have that $|G|=2p$, so there are $2$-Sylow and $p$-Sylow in $G$. $$P\in \text{Syl}_p(G) , \ |P|=p \\ Q\in \text{Syl}_2(G) , \ |Q|=2$$ Let $x\in...

Hey! :o I want to show that each group of order $135$ is nilpotent. We have that $G$ is called nilpotent iff there is a series of normal subgroups $$1\leq N_1\leq N_2\leq \dots \leq N_k=G$$ such that $N_{i+1}/N_i\subseteq Z(G/N_i)\Leftrightarrow [G,N_{i+1}]\subseteq N_i$, right? (Wondering)...

Homework Statement Show that the group (\mathbb Z_4,_{+4}) is isomorphic to (\langle i\rangle,\cdot)? Homework Equations -Group isomorphism The Attempt at a Solution Let \mathbb Z_4=\{0,1,2,3\}. (\mathbb Z_4,_{+4}) can be represented using Cayley's table: \begin{array}{c|lcr} {_{+4}} & 0 &...