Groups Definition and 867 Threads

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 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...

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...

Homework Statement Hi all, i have to identify 5 samples (1,2,3 were solids, 4,5 were liquids) by classifying them as 1) Aliphatic or aromatic and 2) Carboxylic acid, amine (primary, secondary, tertiary) or ammonium carboxylate We did a burn test on the solids, tested solubility in water...

Hi! I'm looking for a complete derivation of space groups as Schoenflies did over 100 years ago... Does anybody know where I can find this paper (in English or in German at least): A. Schoenflies Kristallsysteme und Kristallstruktur, Leipzig, 1891 or maybe a book where the whole process of...

Homework Statement Show that the set of all 2x2 matrices of rank 1 is a submanifold of R^4 Homework Equations The Attempt at a Solution The hint in the book was to show that the determinant function is a submersion on the manifold of nonzero 2x2 matrix M(2) - 0. This is easy to...

Z(G) = { x in G : xg=gx for all g in G } (center of a group G) C(g) = { x in G : xg=gx } (centralizer of g in G) I have to show both are subgroups, but what's the difference in the methods? To me the first set is saying all the elements x1, x2,... in G when composed with every element in g...

D(g) is a representaiton of a group denoted by g. The representaion is recucible if it has an invariant subspace, which means that the action of any D(g) on any vector in the subspace is still in the subspace. In terms of a projection operator P onto the subspace this condition can be written...

Hi there, I 'm currently reading topics relating to type IIB superstring theory. One of the things I am always confused with is Groups. I looked on various websites including Wikipedia but I still haven't quite got it. Could anyone please give me a nice introduction about Groups? What are...

Homework Statement Show that every element of the quotient group \mathbb{Q}/\mathbb{Z} has finite order but that only the identity element of \mathbb{R}/\mathbb{Q} has finite order. The Attempt at a Solution The first part of the question I solved. Since each element of...

As i understand it, the commutation rules for the quantum angular momentum operator in x, y, and z (e.g. Lz = x dy - ydx and all cyclic permutations) are the same as the lie algebras for O3 and SU2. I'm not entirely clear on what the implications of this are. So I can think of Lz as generating...

Hey guys, I've been doing a lot of reading on quantum mechanics lately and realized immediately that i am not going to get far without first understanding the meanings of lie groups, SU groups etc. Now I've loked at wiki but unfortunately wiki is not a very good tool for learning math, it's more...

I am trying to show show that there is no homomorphism from Zp1 to Zp2. if p1 and p2 are different prime numbers. (Zp1 and Zp2 represent cyclic groups with addition mod p1 and p2 respectively). I am not sure how to do this but here are some thoughts; For there to be a homomorphism we...

Homework Statement How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)? The Attempt at a Solution I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had...

Homework Statement Let Cn be a cyclic group of order n. A. How many sub-groups of order 4 there are in C2xC4... explain. B. How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime? explain. C. Prove that if H is cyclic of order 8 then Aut(H) is a non-cyclic group. WHAT is...

I am looking for results which provides the homology and homotopy groups from some property of the space. For instance, if a space X is contractible then H_0(X)=\mathbb{Z} and H_n(X)=0 if n\neq 0. Another example is the Eilenberg MacLane spaces K(\pi,n) where \pi_n(K(\pi,n))=\pi and...

Homework Statement Let P be a p-sylow sbgrp of a finite group G. N(P) will be the normalizer of P in G. The quotient group N(P)/P is cyclic from order n. PROVE that there is an element a in N(P) from order n and that every element such as a represnts a generator of the quotient group...

Homework Statement a) Classify all groups of order 51. b) Classify all groups of order 39. Homework Equations Sylow theorems. The Attempt at a Solution a) C51 b) Z3 X Z13 and Z13 x Z3, the semi-direct product with presentation <a,b|a13=b3=1, ab=a3 > Are these all of the...

Homework Statement What functional groups are present based on the compound's names? A. Methyl Hydroxybenzoate B. 2-Hydroxypropanoic acidHomework Equations The Attempt at a Solution We've learned about the basic Hydrocarbon derivatives in class, but only dealing with problems like...

Homework Statement The problem is: Let G be a group of order 12 ( o(G)=12). Let's assume that G has a normal sub-group of order 3 and let a be her generator ( <a>=G ). In the previous parts of the questions I've proved that: 1. a has 2 different conjucates in G and o(N(a))=6 or...

Homework Statement If the order of G is p^2 and p is prime, then show that G is either cyclic or isomorphic to ZpXZp... Homework Equations The Attempt at a Solution Any hints here will help!

Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products." http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html So, if I've...

Hello Lately, I have been studying some group theory. On my own, I should add, so I don't really have any professor (or other knowledgeable person for that matter) to ask when a problem arises; which is why I am here. I had set out to find all small groups (up to order 30 or something), up to...

Hi guys just a quick question on how I would go about showing the units of Z21 is isomorphic toC2*C6(cyclic groups).I have done out the multiplicative table but they seem to be different to me. What else can I do?

H,K are normal subgroups of a (finite) group G, and K is also normal in H. If G/K and G/H are simple, does it follow that H=K? I'm almost convinced it does, but I'm having trouble proving it. I mean, the cosets of H partition G and the cosets of K partition G in the same way and on top of that...

HELP! Find all abelian groups (up to isomorphism)! I am really confused on this topic. can you give me an example and explain how you found, pleaseee! for example, when i find abelian group of order 20; |G|=20 i will find all factors and write all of them, Z_20 (Z_10) * (Z_2) (Z_5)*...

Homework Statement Write the multiplication table of C_{6}/C_{3} and identify it as a familiar group. Homework Equations The Attempt at a Solution C_{6}={1,\omega,\omega^2,\omega^3,\omega^4,\omega^5} C3={1,\omega,\omega^2} The cosets are C3 and \omega^3C3 I just need help...

Homework Statement Show that every abelian group of order 70 is cyclic.Homework Equations Cannot use the Fundamental Theorem of Finite Abelian Groups.The Attempt at a Solution I've tried to prove the contrapositive and suppose that it is not cyclic then it cannot be abelian. But that has lead...