Groups Definition and 906 Threads

Hi, a quick question: If f is a degree n irreducible polynomial in Q[x] and the Galois group G of f is abelian, then 1. How do we know that G has exactly n elements? 2. Is the Galois group necessary cyclic? I think that since f is irreducible, the Galois group must contain an...

I was studying my brother's old notes to prepare me for my upcoming AP Biology class. I read over six main functional groups (hydroxyl, carbonyl, carboxyl, amino, sulfhydrl, phosphate) and subsequently, there were follow-up questions. One of them asked to draw a molecule with all of the six...

I've been working on this problem and I need just a small hint. Let A and B be solvable subgroups of a group G and suppose that A\triangleleft G . Prove that AB is solvable. My idea: So we have a chain of normal subgroups of A so that their quotient is abelian. We also have a...

Hi, I'm working on some homology problems but I need help figuring out the induced map from a given map, say f:X\rightarrow Y. For example, compute H_* (\mathbb{R}, \mathbb{R}^n - p) where p \in \mathbb{R}^n. So for n=1, we have the long exact sequence 0 \rightarrow...

Let G be a 3-dimensional simply-connected Lie group. Then, G is either 1.)The unit quaternions(diffeomorphic as a manifold to S$^{3}$) with quaternionic multiplication as the group operation. 2.)The universal cover of PSL$\left( 2,\Bbb{R}\right) $ 3.)The...

Homework Statement Show that S_3 has the presentation <x,y|x^3=y^2=(xy)^2=1>Homework Equations x^{-1}=x^2,y^{-1}=y,xyxy=1 xyx=y^{-1}The Attempt at a Solution Let H=<x>, has at most order 3. Then y^{-1}xy=yxy=x^{-1}=x\in < x > x^{-1}xx=x\in < x > so <x>\lhd G Then let <y>=K and use If...

I have seen many types of periodic tables which I gather into 3 categories based on how they handle the Lanthanide series (and the Actinide series is handled in an analogous way). The tables do not say the things I say below, but they imply it by the layout. 1. Lanthanum is in group 3, but not...

Homework Statement Let G be a set with an operation * such that: 1. G is closed under *. 2. * is associative. 3. There exists an element e in G such that e*x = x. 4. Given x in G, there exists a y in G such that y*x = e. Prove that G is a group. Homework Equations I need to prove...

Homework Statement Show that the 5-Sylow subgroups of a group of order 90 is normal. Homework Equations None. The Attempt at a Solution I know that the number \nu_5 of 5-Sylow subgroups must divide 90 and be congruent to 1 mod 5. That means that \nu_5\in\{1,6\}. I also know that...

[b]1. G a Group of order 60, G simple, prove G isomorphic to A5 [b]2. Familiar with Sylow's Theorems, theorems leading up to Sylow. [b]3. We make the assumption that G is not isomorphic to A5 Then "given G cannot have a subgroup of index 2, 3, 4, 5," I can get...

Homework Statement Classify the groups of order 12. Homework Equations None. The Attempt at a Solution The professor has worked this out up to a point. He proved a corollary that states: "Let G be a group of order 12 whose 3-Sylow subgroups are not normal. Then G is isomorphic...

Homework Statement Construct a polynomial of degree 7 with rational coefficients whose Galois group over Q is S7 Homework Equations I need an irreducible polynomial of degree 7 that has exactly 2 nonreal roots. The Attempt at a Solution I have just been using trial and error...

Hello, I came up, and proved, a little theorem. My question is, that do you know this theorem from any other context, or if it has other similar forms (or if it is incorrect). Suppose we have two metric groups A and B, and we know that A is simply connected. If we have a group homomorfism...

Does anyone here know where i can find some information about this groups. Specifically the lie algebra of the generators of these groups?

Homework Statement "Show that the C5 group is not a crystal point group." 2. Relevant information 1) "There exists another type of symmetry operation, called point symmetry, which leaves a point in the structure invariant" 2) "In crystallography, the angle of rotation cannot be arbitrary...

Quotient ring is also know as factor ring but what has it got to do with 'division' in any remote sense whatsoever? I know it is not meant to be division per se but why give the name of this ring the quotient ring or factor ring? What is the motivation behind it? R/I={r in R| r+I} Normally...

Homework Statement Consider the groups G_n= {(a,b) in (C*) * C ; (a,b).(a',b')=( aa', b+(a^n)b' )} where n is in N. Show that if n and m are distinct, then the two groups G_n and G_m are not isomorphic. Ps. In fact G_n = G/ nZ , where G is the group of pairs (t,s) in C * C with group law...

Does anybody know the answer of the following problem? Show that the Lie group of Euclidean motions of R^3 has a Lie algebra g which is perfect i.e., Dg=g but g is not semisimple. By Dg I mean the commutator [g,g] and a semisimple lie algebra is one has no nonzero solvable ideals. Regards

Hi all, Anybody knowes how to find, or at least knows the reference that shows, the real lie algebra of sl(n,H)? By sl(n,H), I mean the elements in Gl(n,H) [i.e. the invertible quaternionic n by n matrices] whose real determinant is one. Many Thanks Asi

Hello, I'm taking a seminar in group theory, and the main focus of the course is the classification of low-order groups (by classification, I mean finding all of the groups of a given order). I've got the hang of classification of (some) groups G that are products of powers of 2 distinct...

Describe al group homomorphisms \phi : C_4 --> C_6 The book I study from seems to pass over Group Homomorphisms very fast. So I decided to look at Artin's to help and it uses the same definition. So I think I am just not digesting something I should be. I know it's defined as \phi (a*b)...

I read on wiki* that a (pointed) topological space is simply connected iff its fundamental group is trivial. But I don't see how this in accordance with the R² caracterisation that U is simply connected iff it is path-connected and has no holes in it. Take the closed unit-disk with a point of...

Hey guys. I've been stuck on the following thing for a little while now. Some help would be appreciated. If p divides o(G) (G an abelian group and p a prime), then show that G(p) = {g from G | o(g) = p^k for some k } I keep going round in circles. P.S. - this is not a homework...

I know of the "result" that if two pointed spaces are homeomorphic, then the group homomorphism induced by such an homeomorphism if actually an isomorphism between the fundamental groups of these pointed spaces. But is there a link between the fundamental groups of homotopy equivalent spaces?

I understand that E^2 - B^2 is invariant under various transformations. If we consider the vector ( E, B ) as a column, then E^2 - B^2 is preserved after mutiplication by a matrix - | cosh( v) i.sinh(v) | | i.sinh(v) cosh(v) | I think this transformation belongs to a group...

Show that whenever ab = ba, you have ba^(-1) = a^(-1)b. I don't know how to slove problem. pls help me..

How do I proof that groups of an even order must have an element of order 2? I have a vague idea, but I don't know how to put my idea together. Aside from identity, there are an odd number of elements in my group. So one element will not have a partner and will have to be multiplied by itself...

I hear all the time people saying things like "groups are the algebraic equivalent to the notion of symmetry". Yet I don't think I've ever really understood what was meant by this. I'm familiar with groups of symmetries (like the dihedral groups), but in these cases, to me it just seems like...

\mathbb{R}^3 has an associative multiplication \mu:\mathbb{R}^3\times \mathbb{R}^3 \rightarrow \mathbb{R}^3 given by \mu((x,y,z),(x',y',z'))=(x+x', y+y', z+z'+xy'-yx') Determine an identity and inverse so that this forms a Lie group. Well, clearly e=(0,0,0) and the inverse element is...

I am currently living in Colorado Springs, CO and am debating whether or not I want to go back to school to earn a second undergraduate degree in physics and then hopefully go on to earn a Ph.D. I started out in college as a physics major but switched to behavioral science. I am hoping to...

Hi, Ok, I have notice that for several finite groups the following situation occurs... I will use the non-abelian group of order 27 to illustrate the point I'm making: The group has 11 charachers, 9 of which are linear. The group has derived subgroup G' (= Z(G) the centre of the...

Seems to be isomorphic to Z, but I can't seem to be able to prove it. Am I right? If I am, how do I prove it?

According to my notes on SUSY 'as everyone knows, every continuous group defines a manifold', via \Lambda : G \to \mathcal{M}_{G} \{ g = e^{i\alpha_{a}T^{a}} \} \to \{ \alpha_{a} \} It gives the examples of U(1) having the manifold \mathcal{M}_{U(1)} = S^{1} and SU(2) has...

Hi, I'd appreciate help with this problem; Person X and Y have equal number of kids. There are 3 movie tickets. The probability that 2 tickets go to kids of one and 1 ticket goes to the kids of other is 6/7. How many kids do X and Y have? Thanx

Hi folks. I've come across a method to determine the controllability of a quantum system that depends on the Lie group generated by the commutator of the skew-Hermetian versions of the field free and interaction (dipole) Hamiltonians. If, for an N dimensional system the dimension of the group...

Here is the problem concerning permutation groups: u = 1 2 3 4 ------- 3 4 2 1 Show that there is no p such that p^2 (the second permutation) = uI've tried just substituting values for p1, p2, p3 and p4 in: 1 2 3 4...

It seems that I am having difficulty understanding some of the definitions given to me concerning Free Groups. I just finished a thread on free groups where I learned that a free group is one that is constructed from an underlying basis set that is enlarged so that one may form all finite words...

I'm not sure where to start on this one at all, very confused. I don't want anyone to do the entire problem for me just point me in the right direction. I know how to compute probability from simple random events but this question just confuses the heck out of me :( Question: A labor...

The only definition of a free group I have is this: If F is a free group then it must have a subgroup in which every element of F can be written in a unique way as a product of finitely many elements of S and their inverses. Now, is it possible to form a commutator subgroup of F? That is...

U = e^{\frac{1}{2} B} = \cos(\frac{1}{2} \theta) + b \sin(\frac{1}{2} \theta) we can then write: U = e^{\frac{1}{2} \theta b} = \cos(\frac{1}{2} \theta) + b \sin(\frac{1}{2} \theta) And if we rely on Joe's expression, r=\frac{\theta}{2} (rotor angle is always half the rotation): U = e^{br}...

This is question 53\gamma. Given a group G of transformations that acts on X... and a subgroup of G, Go (g * x = x for all x for each g in Go), show that the quotient group G/Go acts effectively on X. A group G "acts effectively" on X, if g * x = x for all x implies that g = e, where g is a...

First post here...so...Hi all. :) Is there any good explanation for why alkyl groups donate electron density? Tried google, tried this site's search function...nada. Books I have access to just state it when explaining eg tertiary carbocation stability, without offering explanation.

Prove all groups of order 99 are abelian: I'm stuck right now on this proof, here's what I have so far. proof: Let G be a group such that |G| = 99, and let Z(G) be the center of G. Z(G) is a normal subgroup of G and |Z(G)| must be 1,3,9,11,33, or 99. Throughout I will make repeated...

Hate crimes, "groups", whatever Yet another Larry David inspired thought here... Who exactly gets to define what is a group and what isn't a group and what groups can be defined as targets for "hate crimes"? On Curb your Enthusiasm, Larry David was trying to convince a police officer that a...

Dear Friends, I have many questions about the special Orthogonal Group SO(n) and the Special Unitary Group SU(n). The first, SO(n) has \frac {n (n-1)}{n} parameters or degrees of freedom, and the second, SU(n) has n^2 -1. If I take for example the group SO(3), this has 3 degrees of...

Hello, I'm not sure if the following definitions of "properly discontinuous" (=:ED) are equivalent. I found them in two different books: 1) G: group M: top. manifold G is ED \Leftrightarrow for all compact K \in M there only finitely g_i \in G exist with g_i(K) \cap K \neq \emptyset...

I just had an exam, and I'm curious to see if I got it right, because my professor didn't do it like I did, and I didn't have time to hear his final answer. Well, you tell me if I have any mistakes: Let G be such a group. 147=3*7^2. Let P be the 7-Sylow subgroup (it is unique because it is...

Would it be possible to infer that b^5 = e (where e is the group's identity element) from b^{5} a = ab^{5} given that a^{2}=e? (Basically we are given b^{2}a=ab^{3} and a^{2}=e and asked to show that b^{5}=e, though I've managed to infer the "equation" above and I can't quite see how...

Let (G,*) be a group and (S,*) a subgroup of G. Prove that if for an element a in G, there exists m,n in Z, which are relatively prime, such that a^m and a^n is in S, then a is in S. At the moment, I think the problem is trivial but something just tells me it is not.

I'm really stuck on these two questions, please help! 1. Let G={invertible upper-triangular 2x2 matrices} H={invertbile diagonal matrices} K={upper-triangular matrices with diagonal entries 1} We are supposed to determine if G is isomorphic to the product of H and K. I have concluded...