Groups Definition and 906 Threads

This particular problem has me stumped. I've looked through all the equations given in the book, but none seem to fit this problem correctly. Instead, I tried solving it logically, but I'm stuck on what seems to be the last step of every part. Any help is appreciated! Homework Statement...

Homework Statement Let A be a normal subgroup of a group G, with A cyclic and G/A nonabelian simple. Prove that Z(G)= AHomework Equations Z(G) = A <=> CG(G) = A = {a in G: ag = ga for all g in G} My professor's hint was "what is G/CG(A)?" The Attempt at a Solution A is cyclic => A is...

Homework Statement Does every group whose order is a power of a prime p contain an element of order p? Homework Equations The Attempt at a Solution I know it certainly can contain an element of order p. I also feel that |G|=|H|[G:H] might be useful. Any help is appreciated!

I know full well the proof using Lagrange's thm. But is there a direct way to do this without using the fact that the order of an element divides the order of the group? I was thinking there might be a way to set up an isomorphism directly between G and Z/pZ. Clearly all non-zero elements...

This is what you get if you compile numerous attractive females and then take all their features to come up with archetypal faces that is supposed to best represent the group they belong to.

Homework Statement [PLAIN]http://img541.imageshack.us/img541/9880/34132542.gif The Attempt at a Solution For part (a), I think since the order of an element g is the smallest integer n such that gn=e, we will have: 8n mod 65 = 1 => n=4 64n mod 65 = 1 => n=2 14n mod 65 =1 =>...

I'm currently learning Lie groups/algebras and I am trying to find the infinitesimal generators of the special orthogonal group SO(n). It is the n-dimensions that throws me off. I know that the answer is n(n-1)/2 generators of the form, X_{\rho,\sigma}=-i\left(x_{\rho}\frac{\partial}{\partial...

Homework Statement [PLAIN]http://img641.imageshack.us/img641/8448/63794724.gif The Attempt at a Solution Firstly, how do I list the elements of H? According to Lagrange's theorem if H is a subgroup of G then the number of distinct left (right) cosets of H in G is |G|\|H|. So I...

Homework Statement Let [G,+,0] be a non-abelian group with a binary operation + and a zero element 0 . To prove that if both the zero element and the inverse element act on the same side, then they both act the other way around, that is: If \forall a \in G , a + 0 = a , and a + (-a)...

Homework Statement Suppose (G, \circ) is a group. Define an operation \star on G by a \star b = b \circ a for all a,b \in G. Show that (G,\star) is a group. The Attempt at a Solution So, I have to show that (G,\star) satisfies the associativity, identity, inverse and closure...

Suppose \mathfrak{g} and \mathfrak{h} are some Lie algebras, and G=\exp(\mathfrak{g}) and H=\exp(\mathfrak{h}) are Lie groups. If \phi:\mathfrak{g}\to\mathfrak{h} is a Lie algebra homomorphism, and if \Phi is defined as follows: \Phi:G\to H,\quad \Phi(\exp(A))=\exp(\phi(A))...

Homework Statement I am trying to solve a question from Abstract Algebra by Hernstein. Can anyone give me hint regarding the following: Show that a group can not be written as union of 2 (proper) subgroups although it is possible to express it as union of 3 subgroups? Thanks...

Is there a relationship between the homotopy groups of a pair (X,A) and of the quotient X/A ? It feels like they should be equal under mild hypothesis. More precisely, I am interested in the case where X is a smooth manifold and A a submanifold. Thx

Why are there so few research groups that focus on relativity and gravity in the US? It may be only my perception, but it seems that studying general relativity is rather "standard" in the UK. The schools I've looked at all seem to have a few classes on relativity at the undergraduate level...

Homework Statement This is a problem from a chapter entitled "Permutation Groups" of an abstract algebra text. 1. Let α = ( 1 3 5 7 ) and β = (2 4 8) o (1 3 6) ∈ S8 Find α o β o α-1. 2. Let α = ( 1 3) o (5 8) and β = (2 3 6 7) ∈ S8 Find α o β o α-1. Homework Equations Sn is the set...

Hi, Everyone: LetB: E-->X be a line bundle, with scructure group G and X has a CW -decomposition. I am trying to understand why/how, if the structure group G of B is connected, then any trivialization over the 0-skeleton of X can be extended to a trivialization of the...

If the Cayley table, of a finite set G, is a latin square (that is, any element g appears once and only once in a given row or column) does it follow G is a group? I know the converse is true, and it seems reasonable that this is true. Since the array will be of size |G|x|G|, inverses exist and...

Hi. I'm looking for an introductory book on Lie Groups and Lie Algebras and their applications in physics. Preferably the kind of book that emphasizes understanding, applications and examples, rather than proofs. Any suggestions? Edit: Please move this to Science Book Discussion.

I would like to know how much weightage is taken into account for activities outside class, when applying for a university scholarship... Are sports activities given preference over, for example, science interest groups such as astronomy? What advantage does multiple club memberships hold? Is...

Alright, I understand that there are redundant degrees of freedom in the Lagrangian, and because transformations between these possible "gauges" can be parametrized by a continuous variable, we can form a Lie Group. What I am not so firm upon is how Lie Algebras, specifically, the Lie Algebra...

Hi Everyone, Back in college i informally learned what i would call point group theory. Most of it never touched on continuous transformations. When I learned it back then it was all pretty straight forward. Recently I have been trying to learn about Lee groups (to understand symmetries in...

A while ago I heard the following two facts about semi-simple Lie groups (though I have a feeling they may have to be restricted to connected semi-simple Lie groups): 1. That semi-simple Lie groups are classified by their weight (and co-weight) and root (and co-root) lattices; 2. That all of...

Hi All, I am trying to learn about Linear Algebraic Groups. I am using the book by James Humpreys. I love the subject, but I find it a bit, say, not so beginner-friendly. My goal is not being spoon-fed, but I am very interested in finding a source(s) whereby one is able to go through some...

Is it true that cyclic groups with x^n = 1 the only finite groups (with order n)? I've been experimenting with a few groups and I think this is true but I'm not sure.thanks

Homework Statement I will present a vary simplified version of the problem I am trying to model. Essentially I took 100 people and split them into 10 groups. To each group I tried to sell them a product. To the first group I priced the product at $5, to the second I priced it at $15, to the...

I have 2 questions: 1. Can anything be said about the order of a group from the order of its generators (or vice versa)? E.g. if a group G = <a,b>, is there any theorem that says the order of elements a, b is divisible by the order of G, or maybe, if G = <a,b>, then the order of G is the...

I have a few question concerning factor groups. 1. In a proof for the fact that if a finite factor group G/N has 2 elements, then N is a normal subgroup of G, it says: "For each a in G but not in H, both the left coset aH, and right coset Ha, must consist of all elements in G that are not in...

My wife and I support the World Wildlife Fund, the National Wildlife Federation, and the Arbor Day Foundation to try to preserve habitat, fund research and repopulation efforts, and encourage tree-growth. We have a life-time supply of return address labels and little note-pads, though I wish...

I have not seen why SU(2) and SO(3) groups are isomorphic?

Homework Statement Let G be a finite group and N\triangleleftG such that |N| = n, and gcd(n,[G:N]) = 1. Proof that if x^{n} = e then x\inN. Homework Equations none. The Attempt at a Solution I defined |G| = m and and tried to find an integer which divides both n and m/n. I went for some X...

so factor groups/quotient groups have been tripping me up recently and if i could a definition and maybe an example from you guys that would help me out a lot.

As a way to keep busy in between semesters I decided to work my way through Algebra by Dummit and Foote in order to prepare for the fall. Working my way through quotient groups is proving to be quite difficult and as a result I'm stuck on an exercise that looks simple, but I just don't know...

Hi folks - I have a couple of basic questions about fundamental representations. First of all, does every group have a set of fundamental representations? Secondly, I know that in the case of the (compact?) semi-simple groups, any other representation of the group can be constructed by...

Hello! I'm currently taking a course in group- and ring theory, and we are now dealing with a chapter on finitely generated modules over PIDs. I have stumbled across some problems that I can't really get my head around. It is one in particular that I would very much like to understand, and I...

I'm just having a little trouble getting my head around how representation theory works. Say for example we are working with the dihedral group D8. Then the degrees of irreducible representations over C are 1,1,1,1,2. So there are 4 (non-equivalent) irreduible representations of degree 1...

My abstract algebra book is talking about reducing the calculations involved in determining whether a subgroup is normal. It says: If N is a subgroup of a group G, then N is normal iff for all g in G, gN(g^-1) [the conjugate of N by g] = N. If one has a set of generators for N, it suffices...

Help with permutation groups... How do i show that if P: G1 --> G2 is a group homomorphism, then the image, P(G1) = {g belongs to G2 , s.t. there exists h belonging to G1 , P(h) = g}, is a subgroup of G2 Also if we let G be a group, and Perm(G) be the permutation group of G. How do i show...

Homework Statement View S^3 as the unit sphere in C^2. Now, 1. What are the path connected components of the subset of S^3 described by the equation x^3 + y^6 = 0, where the x and y refer to the coordinates (in C)? 2. Is it true that the similar subset x^2 + y^5 = 0 is homeomorphic to...

Homework Statement The question is, "How many conjugates does (1,2,3,4) have in S7? Another similar one -- how many does (1,2,3) have in S5? The Attempt at a Solution I know that the conjugates are all the elements with the same cycle structure, so for (123) I found there are 20...

1: Is a group of permutations basically the same as a group of functions? As far as I know, they have the same properties: associativity, identity function, and inverses. 2: I don't understand how you convert cyclic groups into product of disjoint cycles. A cyclic group (a b c d ... z) := a->b...

Homework Statement Any help with this question would be great: G is a group such that |G| = pk, p is prime and k is a positive integer. Show that G must have an element of order p. The hint is to consider a non-trivial subgroup of minimal order. Homework Equations Lagrange...

Homework Statement Consider the cyclic group Cn = <g> of order n and let H=<gm> where m|n. How many distinct H cosets are there? Describe these cosets explicitly. Homework Equations Lagrange's Theorem: |G| = |H| x number of distinct H cosets The Attempt at a Solution |G| = n...

Homework Statement Let G be the group { \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix} | a, b, c are in Z_p with p a prime} Then let K = { \begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix} | b in Z_p} The map P: G --> Z*p x Z*p is defined by P(...

I have a pretty basic question about direct sum/product of groups. Say you were given the group (Z4 x Z2, +mod2). Now I know that Z4 x Z2 is given by { (0,0), (0,1), (1,0), (1,1), (2,0), (2,1), (3,0), (3,1) }. So now if you were going to add together two of the elements using the binary...

I need to calculate the fundamental group of the following spaces: X_1 = \{ (x,y,z) \in \mathbb{R}^3 | x>0 \} X_2 = \{ (x,y,z) \in \mathbb{R}^3 | x \neq 0 \} X_3 = \{ (x,y,z) \in \mathbb{R}^3 | (x,y,z) \neq (0,0,0) \} X_4 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0,y=0...

Homework Statement Let G1 and G2 be groups, let G = G1 x G2 and define the binary operation on G by (a1,a2)(b1,b2):=(a1b1,a2b2) Prove that this makes G into a group. Prove G is abelian iff G1 and G2 are abelian. Hence or otherwise give examples of a non-cyclic abelian group of order 8...

X1 = {(x; y; z) ∈ R^3 | x > 0} just need to check my thinking is pi1(X1) = {1} i.e. trivial

Homework Statement Let a,b be elements of a group G. show that if ab has finite order, then ba has finite order. Homework Equations The Attempt at a Solution provided proof: Let n be the order of ab so that (ab)n = e. Multiplying this equation on the left by b and on the right by...

Hi, F is a finite field. The problem is set up as follows: Let V be a 2-dimensional vector space over F. Let Omega=set of all 1-dimensional subspaces of V. I've constructed PGL(2,F) by taking the quotient of GL(2,F) and the kernel of the action of GL(2,F) on Omega. Similarly for PSL(2,F)...

Homework Statement Let G be a group with a finite number of elements. Show that for any a in G, there exists an n in Z+ such that an=e.Homework Equations a hint is given: consider e, a, a2,...am, where m is the number of elements in G, and use the cancellation laws.The Attempt at a Solution so...