Groups Definition and 867 Threads

Hi, everyone: I have been looking for a while without success, for the definition of equivalence for unimodular quadratic forms defined on Abelian groups . I have found instead ,t the def. of equivalence in the more common case where the two forms Q,Q' are defined on vector...

Hi everyone! I would like to solve some questions: Classify up to isomorphism the four-sheeted normal coverings of a wedge of circles. describe them. i tried to to this and it is my understanding that such four sheeted normal coverings have four vertices and there are loops at each of...

If A and B are subsets of G, let A*B denote the set of all points a*b for a in A and b in B. Let A^(-1) denote the set of all points a^(-1), for a in A. a)A neighborhood V of the identity element e is said to be symmetric if V = V^(-1) . If U is a neighborhood of e, show there is a...

Homework Statement I am interested in proving that A5 has no normal subgroups except itself and {e}. The Attempt at a Solution Some proofs that I have seen use centralizers to do this, but since I haven't gone through that yet I think there should be some say to do it without them. My...

Let n be a positive integer and let m be a factor of 2n. Show that Dn (the dihedral group) contains a subgroup of order m. I'm not really sure where to start with this one. I know that Dn is generated by two types of rotations: flipping the n-gon over about an axis, and rotating it 2π/n...

Let An (the alternating group on n elements) consist of the set of all even permutations in Sn. Prove that An is indeed a subgroup of Sn and that it has index two in Sn and has order n!/2. First of all, I need clarification on the definition of an alternating group. My book wasn't really good...

Homework Statement Partition the following collection of groups into subcollections of isomorphic groups. a * superscript means all nonzero elements of the set. integers under addition S_{2} S_{6} integer_{6} integer_{2} real^{*} under multiplication real^{+} under multiplication...

Homework Statement Recall that Vm is the set of all invertibles in Z/m a) List the elements in V15 b) Find all the subgroups of V15 c) Is V15 cyclic? why? Homework Equations The Attempt at a Solution From my notes: a) V15 = {1, 2, 4, 7, 8, 11, 13, 14} b)...

Homework Statement Let H be the subgroup of GL(3, \mathbb{Z}_3) consisting of all matrices of the form \left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right], where a,b,c \in \mathbb{Z}_3. I have to prove that Z(H) is isomorphic to \mathbb{Z}_3 and that H/Z(H) is...

Is there a formula for determining the number of different groups up to isomorphism for a group of a given order?

Hello, What about gauging discrete groups ? (C, P, T (??), Flavour Groups, Fermionic number symmetry...)

Is it true that any group of order 30 has a normal (hence unique) Sylow-5 subgroup? I know that that the only possibilities for n(5) are 1 or 6. Now suppose there are 6 sylow 5 subgroups in G. This would yield (5-1)6=24 distinct elements of order 5 in G. Now there is only 30-24=6 elements left...

Homework Statement Let f : G → H be a homomorphism of Abelian groups. 1. Show that f (0) = 0. 2. Show that f (−x) = −f (x) for each x ∈ G. Homework Equations The Attempt at a Solution My background in topology / group theory is next to nothing. 1. Show that f(0) = 0. My attempt is as...

Do they exist? What are some examples? Are there any applications? What are some good books on the topic?

Let H be normal in G, H cyclic. Show any subgroup K of H is normal in G. I was thinking about using the fact that subgroups of cyclic groups are cyclic, and that subgroups of cyclic groups are (fully)Characteristic (is that true?). Then we would have K char in H and H normal in G. Hence K...

I have two questions, they aren't homework questions but I figured this would be the best place to post them (they are for studying for my exam). Homework Statement How many elements of S_6 have order 4? Do any elements have order greater than 7? Homework Equations S_6 is the...

I've taken a course in Linear Algebra, so I'm used to working with vector spaces. But now, I'm reading Griffith's Introduction to Elementary Particles, and it talks about groups having closure, an identity, an inverse, and being associative. With the exception of commutativity (unless the...

How many symmetries (and what symmetries) and how many elements do the transformation groups of the equilateral triangle and the icosahedron have? thanks

Homework Statement Suppose G is a non-abelian group of order 12 in which there are exactly two elements of order 6 and exactly 7 elements of order 2. Show that G is isomorphic to the dihedral group D12. Homework Equations The Attempt at a Solution My attempt (and what is listed...

I spend much time to study a theorem - Let G be a nonzero free abelian group of finite rank n, and let K be a nonzero subgroup of G. Then K is free abelian of rank s smaller or equal to n. Furthermore, there exists a basis (x1,x2,...,xn) for G and positive integers d1,d2,...,ds where di divides...

i got two groups of numbers "a" and "b" what is the meaning of a\b and b\a ??

Homework Statement Prove that a discrete group G cosisting of rotations about the origin is cyclic and is generated by \rho_{\theta} where \theta is the smallest angle of rotation in G The Attempt at a Solution since G is by definition a discrete group we know that if \rho is a...

Ok, here is something i thought i understood, but it turns out i am having difficulties fully grasping/proving it. Let \theta:G->G' be an isomorphism between G and G', where o(G)=m=o(G'), and both G and G' are cyclic, i.e. G=[a] and G'=[b] So my question is, when we want to find the...

Given H,K and general finite subgroups of G, ord(HK) = [(ord(H))(ord(K))] / ord(H intersection K) I know by the first isomorphism theorem that Isomorphic groups have the same order, but the left hand side of the equation is not a group is it? I am struggling to show this.

How do we know that half of permutations are odd and half are even? Why not 1/4, 3/4 or other proportions?

1. Show that every element of the quotient group G = Q/Z has finite order. Does G have finite order? he problem statement, all variables and given/known data [b]2. This is the proof The cosets that make up Q/Z have the form Z + q, where q belongs to Q. For example, there is a...

is the set of nxn traceless hermitian matrices under addition a group? is the set of nxn traceless hermitian matrices under multiplication a group? is the set of nxn traceless non-hermitian matrices under addition a group? question 1-I thought that traceless means trace=0 is this right...

[b] Show that any group of even order has at least an element of order 2 Homework Equations [b]3. I know that the order of a groups tells you how many elements the group consist, but just randomly assuming that it has at least an order of 2 is what I can't really understand. For example...

While having a discussion with my students this week, a topic came up that I found interesting, and was wondering what other people's views were on it. We were talking about patient compliance with treatment (for diabetes in this case) and the role and availability of support groups. The...

Homework Statement Let G and G' be finite groups whose orders have no common factor. Prove that the homomorphism \varphi G \rightarrow G' is the trivial one \varphi (x) =1 for all x. The Attempt at a Solution My thoughts are that we need to use lagrange's thm. somehow. or maybe...

I'm currently studying factor groups in abstract algebra and needed some help understanding how to determine the order of an element in a factor group Suppose I have Z (mod 12) / <4>. And I choose some random element from Z (mod 12) such as 5 or 7. How would I go about determining the order...

Could someone please give me an example of an ether and an ester. In words, not drawing them. Thank you!

Hi. I'm currently working on a expository paper about quantum groups and Hopf algebras. However, from all the books I've read, they are more about examples (deformations of various groups) than actual interesting results (Or perhaps I just don't understand them enough to draw any interesting...

How can I show that all Non Abelian Groups of order 6 are isomorphic to S_3 without using Sylow's Theorems? I have shown the following: G has a non-normal group of normal subgroup of order 2 The elements of G look like: 1, a, a^2, b, c, d, where a,a^2 have order 3 and others have order 2...

Homework Statement I was curious to know, say we have two even permutations taken out of A_4, say (12)(34) and (123), and we want to find the smallest subgroup of A_4 that contains both these permutations, then how would we go about it. This subgroup in this case will defenitely be A_4...

[b]1. Let G be a Group, and let H be a subgroup of G. Define the normalizer of H in G to be the set NG(H)= the set of g in G such that gHg-1=H. a) Prove Ng(H) is a subgroup of G b) In each of the part (i) to (ii) show that the specified group G and subgroup H of G, CG(H)=H, and NG(H)=G...

Homework Statement Let G be a group and let H,K be subgroups of G. Assume that H and K are Abelian. Let L=(H-union-K) be the subgroup of G generated by the set H-union-K. Show that H-intersect-K is a normal subgroup of L. The Attempt at a Solution How do i start this?

If two groups A and B are isomorphic then by studying one of them, we can deduce all algebraic information about the other? Hence studying one is equivalent to studying the other?

Homework Statement If H, K are subgroups of G, show that H intersect K is a subgroup of H Homework Equations I know that H intersect K is a subgroup of G; I proved this already but I'm wondering how H intersect K is a subgroup of H The Attempt at a Solution I'm quite sure this is true...

Please, help me with the following questions or recommend some good books. 1) We have a simply-connected nilpotent Lie group G and a lattice H in G. Let L be a Lie algebra of G. There is a one to one correspondence between L and G via exp and log maps. a) Is it true, that to an ideal in...

hey guys, I am a first year physics student but my physics lecturer invited me to sit in during her third year physics lecture. Of course i didnt fully understand some of it, but i think i at least grasped the concept of confinement (the lecture was on quantum chromodynamics by the way)...

what is the order of normalizer of Sylow p-subgroup of simple groups A_n?

What are some properties apart from the actual names of the elements that differ between isomorphic groups?

I'm currently going through Hungerford's book "Algebra", and the first proof I found a bit confusing is the proof of the theorem which states that every infinite cyclic group is isomorphic to the group of integers (the other part of the theorem states that every finite cyclic group of order m is...

we all know that set of rationals i a subgroup of set of reals. my question is whether there exsts a group between these tw groups. f yes what it can be? and if no, how to prve the non-existence?

I'm a homeless transient. One constant I've noticed is that around the downtown area in all medium sized ( population 100,000 + ) cities or larger, there are large groups of teenagers that hang around at night, goofing off and shooting the bull for hours. Don't they have any work to do? Don't...

Just a quick question here: I was going through my notes and I noticed that the generators of both these groups are labeled two indices. I was wondering if there is any particular reason for this, since it seems to me that one index would work perfectly well. Thanks

What is the order of S4, the symmetric group on 4 elements? Compute these products in order of S4: [3124] o [3214], [4321] o [3124], [1432] o [1432]. Can I get help on how to do this. The solution's manual gives the answer on how to do the last two, but I don't understand the process...

Homework Statement Prove that if a group G has no subgroup other than G and {e}, then G is cyclic... Homework Equations The Attempt at a Solution we could say that, let a E G - {e} then we construct <a>...

Let H, K be finite groups. Instead of asking what groups G there are such that K can be embedded as a normal subroup and G/K is isomorphic with H (the usual extension problem for groups), I've been thinking about the following: Which groups G exist such that H and K can be embedded as (not...