Groups Definition and 867 Threads

Here is a nice question I know that exponentiating elements of a Lie-Algebra gives you back an element of the Lie-Group. These Lie-algebra-elements generate the Lie-Group transformations. Like the Galilei-group, these Lie-groups are used in theoretical fysics as the great START, I mean they...

My club has five cultural groups. They are literary, dramatic, musical, dancing and painting groups. The literary group meets every other day, the dramatic every third day, the musical every fourth day, the dancing every fifth day and painting every sixth day. The five groups met on the new...

How would I show that two groups are isomorphic? FOR EXAMPLE: Take the group homomorphism φ : ((0, oo), x) → ((0, oo), x) defined by φ (x) = x² Since φ is taking any element in (0, oo) and operating on it by x, does it map one-to-one and onto to (0, oo)? I...

Okay, I'm really scratching my head here. If an Abelian group A has three generators x,y,z and they are subject to three defining relations, say something like x+y+z=0 x-y-z=0 2x-2y+3z=0 then I can solve for x,y,z and find A as a direct sum of cyclic groups, Z_x + Z_y + Z_z. But...

If I have an abelian group A that is the direct sum of cyclic groups, say A=[tex]C_5 \oplus C_35[\tex], would I be right in saying the annihilator of A (viewed as a Z-module) is generated by (5,35)? If not, how do I find it?

Source: Anderson, Principles of Relativity Physics p. 13, prob. 1.4 "Reparametrize the rotation group by taking, as new infinitesimal parameters, ε1 = ε23, ε2 = ε31, and ε3 = ε12 and calculate the structure constants for these parameters." My assumptions: (1) The εij mentioned in...

Given a group of a certain size, I'm interested in determining whether a non-abelian group of that order exists, and if so how many and what are they up to isomorphism. For example, for |G|=6, the permutation group S_3 is non-abelian. How can I show that any non-abelian group of order 6 is...

Is there any general method to construct the generators of a SU(n) group?

There's been some interesting discussions in this sub-forum on IQ, intelligence, the g Factor, races, hereditability, a nation's wealth, eugenics, and other things. One thing I found particularly interesting in this discussion is the work of Cavalli-Sforza into pre-1492 population groups and...

In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions: Is the gauge group for gravity defined as the group of all possible Weyl tensors on a general 4D Riemann manifold? Is it a subgroup of GL(4)? How do you derive the number of gravitational...

In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions: Is the gauge group for gravity defined as the group of all possible Weyl tensors on a general 4D Riemann manifold? How is this group defined in matrix algebra? Is it a subgroup of GL(4). How do...

Hello, This is my question: (i) Let H and K be subgroups of a group G. Prove that the intersection of H and K is also a subgroup of G. (ii) Give an example, using suitable sugroups of the goup of integers with the operation addition, (Z,+), to show that if H and K are subgroups of a...

[SOLVED] Proof Using Def. of Groups Hello, This is my question: Let G be a group. i) Let x and y be elements of G. Prove that (xy)2 = x2y2 iff xy = yx. (Hint: Use the definition g2 = gg). ii) Using part (i) prove that if g2 = u (the unit element) for all g which is an element of...

Hello, I just started doing groups in my algebra class and I am struggling with the abstraction of it as usual. Here is how my class defines a group: A group is a nonempty set G with a binary operation "o" such that for all of x,y,z which are elements of the group the following holds...

The Eightfold Way categories which groups of particles?

Dear Fellow scientists, I have a small but important problem: I need to know the absorption wavelength of epoxy groups. To be more precise, these epoxy groups are bound to an agarose affinity resin via a -((CH)2)12- Spacer. Does anybody know a particular reference or chemistry book I should...

My first post, about rotation groups.. A result about rotation groups. To me this seems clear, simple and very intuitive, but in all the papers and books I've read on the subject I have never seen it presented. Maybe some of you have seen it, or maybe it is new. It is very simple to state...