Groups Definition and 906 Threads

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...

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...