Groups Definition and 867 Threads

Homework Statement I haven't been assigned these questions, but I'm trying to trudge through them to better understand symmetry. This is for my inorganic class. It's just a series of short questions like: C3 – S56 = ? S4 + i = ? C3 + i = ? Stuff like this. And just looking at the...

Why do I run into trouble if I try to define the set of all groups? I get that defining the set of all sets could lead to paradoxes. But how is it that defining the set of all groups somehow leads to the same kind of problems? If I define the set of all groups as all the ordered pairs (x,y)...

I imagine a matrix group, with multiplication as the composition rule, to always possesses the quality of having centre (I,-I), as I can't see when both elements wouldn't commute with all others. On the other hand, though, a centerless group is defined as having trivial centre, i.e. Z=I (which...

Mathematicians have produced a wide variety of long and complex proofs of the existence of free groups, and there appears to be a strong emphasis upon finding better proofs that involve a variety of techniques. (Examples are http://www.jstor.org/stable/2978086 and "www.jstor.org/stable/2317030"...

Hello. I'm not sure what type of problem this is that I'm trying to solve. Any pointers would be greatly appreciated. Suppose you have a list of numbers and you want to form them into, say, 4 groups such that the sum of each group is, as nearly as possible, equal to the sums of each of the...

Suppose we have a compact Lie group ##G##, and a differentiable function ##f:G_0\to\mathbb{R}## from the identity component of ##G## to the real numbers. I'm looking to maximize the value of this function. Being something of a neophyte at optimization, especially of this kind, I decided to...

I surely am missing something about the notion of connectedness, and I clarify this by means of an example: O(n), the orthogonal group, has two subsets with detO=1 and detO=-1. Now, the maximally connected component of O(n) is SO(n), which is the subgroup with detO=1 including the Identity...

My question is exactly what is stated in the title: Does Cayley's theorem imply that all groups are countable? I don't see how a well defined transitive action of an uncountable group on itself. How could you possibly find a set of permutations sending a single element to every other element in...

Homework Statement Show that ##f(e)=e'## and ##f(g^-1)=f(g)^-1## Homework Equations Homomorphism f(x\cdot y)=f(x)\cdot f(y) The Attempt at a Solution I show the first one. Neutral element is element which satisfied ##e\cdot e=e##. So ##f(e)=f(e\cdot e=e)=f(e)\cdot f(e)## So ##f(e)=e'##. But...

I have a question regarding symmetry groups. I've often heard that the Standard Model is a SU(3) x SU(2) x U(1) theory. From what I understand these groups contain the symmetries under which the Lagrangian function is invariant. If so, what does every one of the 3 groups above contain (what...

Example: ##\mathcal{L}[f(t)*g(t)]=F(s)G(s) ## Is this homomorphism, actually isomorphism of groups? ##\mathcal{L}## is Laplace transform.

What does it mean for a condition to be "open"? E.g. it is said that det(A)≠0 is an open condition for a matrix group. Furthermore, this implies that GL(n) has the same dimensions as the group of all nxn matrices as, and I quote, "the subgroup of matrices with det(A)=0 is a subset of measure...

I am not really sure whether this topic belongs here or not, but since my example will be a certain one I will proceed here... Someone please explain me what "isomorphism" means physically? For example what is the deal in saying that the proper orthochronous Lorentz group is isomorphic to...

In Beachy and Blair: Abstract Algebra, Section 3.8 Cosets, Normal Groups and Factor Groups, Exercise 17 reads as follows: ---------------------------------------------------------------------------------------------------------------------- 17. Compute the factor group ( \mathbb{Z}_6 \times...

I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms. Exercise 17 in Section 3.1 (page 87) reads as follows: ------------------------------------------------------------------------------------------------------------- Let G be the dihedral group od order 16. G = <...

Homework Statement If ##|G| = p^2## where ##p## is a prime, then ##G## is abelian. 2. The attempt at a solution The book my proof gives is confusing at the very last part. Suppose ##|Z(G)| = p##. Let ##a\in G##, ##a\notin Z(G)##. Thus ##|N(a)| > p##, yet by Lagrange ##|N(a)| \ | \ |G| =...

Combinations of groups question [edited question] The camera club has 5 members and the mathematics club has 8. There is only one member common to both clubs. In how many ways could a committee of four people be formed with at least one member from each club? I am confused about the "one...

I would like to know why $M_n$ $\not\cong$ $O_n$ x $T_n$, where $M_n$ is the group of isometries of $\mathbb R^n$, $O_n$ is the group of orthogonal matrices, and $T_n$ is the group of translations in $\mathbb R^n$. **My attempt:** Can I show that one side is abelian, while the other group is...

Assumptions 1) a=absorption 2) f=fission 3) ∅=neutron flux 4) time independent 5) group 1, fast neutrons 6) group 2, thermal neutrons 7) All fission neutron are boring in fast group 8) All neutrons created by thermal group, thus vƩf2 exists vƩf2 does not 9) Down scattering occurs but up...

How would I construct noncyclic groups of whatever order I want? For example g is order 8.

Suppose θ is a differential 1 form defined on a manifold and with values in the Lie algebra of a Lie group,G. On MxG define the 1 form, ad(g)θ ,where θ is extended by letting it be zero on the tangent space to G How do you compute the exterior derivative, dad(g)θ ? BTW: For matrix...

Hey! From MathWorld on solvable group: But why is that a special case? The way I understand it: the normal series can always be made such that all composition factors are simple, but then the composition factors are both simple and Abelian, and hence (isomorphic to) \mathbb Z_p, i.e. the...

Let G be a group and my book defines closure as: For all a,bε G the element a*b is a well defined element of G. Then G is called a group. When they say well defined element does that mean I have to show a*b is well defined and it is a element of the group? Or do I just show a*b is closed under...

1. Which of the following is a group? To find the identity element, which in these problems is an ordered pair (e1, e2) of real numbers, solve the equation (a,b)*(e1, e2)=(a,b) for e1 and e2. 2. (a,b)*(c,d)=(ac-bd,ad+bc), on the set ℝxℝ with the origin deleted. 3. The question...

Let F be a field. R is an element of Mat(2,2) [a -b b a] for a, b in F with matrix operations. a. Show that R is a commutative ring with 1 and the set of diagonal matrices are naturally isomorphic to F . b. For which of the fields Q , R , C , F5 , F7 is R a field? c. Characterize...

Hi all! I am ready for applying for my PhD which I want to be on String Theory and in Europe. I am finishing my master's degree in Theoretical Physics and currently I am researching Holographic Renormalization for asymptotically non-AdS space-times. I am interested in all aspects of String...

Homework Statement H is a subgroup of G, and a and b are elements of G. Show that Ha=Hb iff ab^{-1} \in H . The Attempt at a Solution line 1: Then a=1a=hb for some h in H. then we multiply both sides by b inverse. and we get ab^{-1}=h This is a proof in my book. My question is...

Hello, PF have helped me a lot understanding a lot of important things in physics, I hope you guys can help me with this too :). I have problems understand the symmetry groups. I know there are groups like SU(2), O(3).. etc. But I have no idea how they represent certain particles.So particles...

Homework Statement From Advanced Modern Algebra (Rotman): Definition Let p be a prime. An abelian group G is p-primary if, for each ##a \in G##, there is ##n \geq 1## with ##p^na=0##. If we do not want to specify the prime p, we merely say that ##G## is primary. If ##G## is any abelian...

\renewcommand{\vec}[1]{\mathbf{#1}} Here is an excerpt from the text: "[...]Theorem 12.5 The only finite symmetry groups in ℝ^2 are \mathbb{Z}_n and D_n. PROOF. Any finite symmetry group G in \mathbb{R}^2 must be a finite subgroup of O(2); otherwise, G would have an element in E(2) of...

I am trying to self-learn group theory from an online pdf textbook, but some of the exercises are tough and the answer section doesn't help much (it only says "True"). Homework Statement Prove or disprove: Every abelian group of order divisible by 3 contains a subgroup of order 3.Homework...

Hello everybody! So, I've learned that in a path connected space, all fundamental groups are isomorphic. Indeed, if ##\gamma## is a path from ##x## to ##y##, then we have an isomorphism of groups given by \Phi_\gamma : \pi_1(X,x)\rightarrow \pi_1(X,y): [f]\rightarrow [\overline{\gamma}]\cdot...

Quantum Gravity: "Lie Groups" vs. "Banach Algebras & Spectral Theory" I'm interested in researching quantum gravity & non-commutative geometry. I am planning to take one math course outside of my physics classes this Fall to help, but can't decide between two: "Lie groups" or "Banach algebras &...

So, I have a topological group ##G##. This means that the functions m:G\times G\rightarrow G:(x,y)\rightarrow xy and i:G\rightarrow G:x\rightarrow x^{-1} are continuous. I have a couple of questions that seem mysterious to me. Let's start with this: I've seen a statement...

So I'm trying to understand this paper (found here: http://arxiv.org/abs/1301.3411) but my math skills are very limited. These include: -Groups (the very basics, like the first of Charles Pinter's book) -Analysis (the very basics) But what all books/papers/topics would you suggest I...

It was suggested in Smolin's book "The trouble with physics" that stringy people may have succumb to a group condition called groupthink - this is where social factors lead people to defective decision making processes. More generally, no offense to string people as it might be just a mild...

For another thread https://www.physicsforums.com/showthread.php?p=4420542 , I want to give simple motivations for the definition of two groups being isomorphic and related topics. My explanations have become so lengthy that I decided to post them here in 3 parts. ( They are probably too...

Hey guys, I'm really interested in finding out how to deal with differential equations from the point of view of Lie theory, just sticking to first order, first degree, equations to get the hang of what you're doing. What do I know as regards lie groups? Solving separable equations somehow...

The definition given is... "Let ##\phi: G \rightarrow H## be a homomorphism with kernel ##K##. The quotient group ##G/K## is the group whose elements are the fibers (sets of elements projecting to single elements of H) with group operation defined above: namely if ##X## is the fiber above...

Was wondering if the only required definition for finite groups is closure (maybe associativity as well). It seems that is all that is necessary. The inverse and identity necessarily seem to follow based on the fact that if I multiply any element by itself enough times, I have to repeat back to...

I am familiar with what SO(2) means for example but am unclear what SO(1,1) refers to. This came up in a classical physics video lecture when lie groups were discussed and the significance of the notation was glossed over. Second question: is the dimensionality of such a group the same as the...

EDIT: To moderators, I frequent this forum and the Calculus forum and I have accidently put this in Introductory Physics. Can it be moved? Sorry for inconveniences. Homework Statement 1)If ##G \cong H \times \mathbb{Z}_2, ## show that G contains an element a of order 2 with the property that...

Homework Statement Let: G = { \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \in GL(2,ℝ)} H = { \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \in GL(2,ℝ)} K = { \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} \in GL(2,ℝ)} Is G isomorphic to H x K? Homework Equations...

First let me apologise for the improper use of 'groups'. I'm not a mathematician but I know that 'groups' means something specific. Anyway, here is my problem. The exact circumstances of my problem are esoteric and bothersome to explain, and I don't want to distract you with details that do...

Show that a•b=(a+b)^2 has no identity for real numbers Hi this is a new topic please help Thanks

Hi, All: I've been curious about this for a while: Why are there different names for different groups of animals? We have ( I think) schools of fish,packs of wolves, etc. I imagine it may have to see with the fact that groups of animals are organized differently. Still...

Question about matrix groups and conjugate subgroups? This question concerns the group of matrices L = { (a 0) (c d) : a,c,d ∈ R, ad =/ 0} under matrix multiplication, and its subgroups H = { (p 0, (p - q) q) : p,q ∈ R, pq =/ 0} and K = { (1 0, r 1) : r ∈ R}Show that one of H and K is a normal...

There is a corollary in our textbook that states "Let G be a group of order 12 whose 3-Sylow subgroups are not normal. THen G is isomorphic to A_4." I attached the proof of this corollary and an additional corollary and proposition that was used for the proof. The 2nd last paragraph is a bit...

I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism...

Hello, I am unsure of how to utilize protecting groups in organic synthesis problems. What would be an immediate sign to the trained eye that one should use a protecting group such as TBDMS-Cl?