Groups Definition and 867 Threads

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...

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...

I need to prove that the center of the product of two groups is the product of their centers. If I let G and H be two groups, then from definitions, the center of G is Z(G)={z in G such that zg=gz for g in G} and the center of H is Z(H)={z in H sucht that zh=hz for all h in H}. Also, the...

Hello, I'm studying homology and graded groups have come up. I don't completely understand what they are. Wikipedia didn't have an entry on graded groups, but on graded algebras and other graded stuff and the definitions there seemed different than the way graded groups have been used...

1. Describe all groups G which contain no proper subgroup. This is my answer so far: Let G be a such a group with order n. Then the following describe G: (a) Claim that every element in G must also have order n. Proof of this: If this wasn't true, the elements of lower order (elements of...

Can isomorphism of matrix groups \phi: G_1 \rightarrow G_2 always be expressed by \phi(M) = S M S^{-1}?

how to define R\Q?(under addition) R\Q={a+Q:? <a<?} a€R but if it is not bounded then it will repeat please help me n

Let p and q be primes with q < p, and q ł (p-1). If G is a group with |G| = qp², then there are two possibilities if G is abelian by the fundamental theorem for finitely generated abelian groups: Zp x Zpq and Zqp² G has a normal Sylow-p subgroup (of order p²) and if we call it H, and call...

I am not really clear on what is meant by commutators. I know that the commutator of G is ABA^-1B^-1, but I am not sure how to check if a group is solvable by having the commutator eventually equal the trivial group. For example, I know that the Heisenberg group of 3x3 upper triangular...

Hi, I need some serious help in paradoxical groups! 1) Given vectors v1,v2 in R2 and w1,w2 in R2 (none lieing on a line thru the origin), show that you can find a unique C such that Cv1=w1 and Cv2=w2. 2) Show that a finite group is not very paradoxical. 3) Is F3 paradoxical? Is Z...

group theory : orbits hi. I'm trying to calculate the orbits of some simple groups. I have found many explanations of what they are, but no example calculations. does anyone have any ideas where to look. I'm trying to calculate the orbit of SO(3). thanks

Does anybody have any thoughts on why atoms that are warm need to radiate heat in order to stop vibrating or bond back together? The obvious example is the thermos. That is a thermos has the hot liquid contained in a glass container with a mirrored surface towards the inside and then a...

1. Describe the elements of the groups c2mm, p4mm, and p3m1. My book doesn't do a good job of explaining this notation, any help? 2. Let m and n be positive integers. Prove that there is a homomorphism from the free group generated by n generators, F_n, onto the free group generated by m...

Show that if G is a finite group of even order, then G has an odd number of elements of order 2. I'd appreciate a tip or two. I really don't see how the order of the elements of a group is linked to the order of that group.

I wanted to restart this discussion b/c the previous thread got sidetracked, and something about it has left me deeply confused, and I think my confusion is similar to the original posters confusion. In Wigners theory, particles (like say an electron) are unitary irreducible representations...

Assume G is a finite group and H = \left\{ {g \in G|g^n = e} \right\} for any n>0. e is identity. I have been able to show that if G is cyclic, then H has at most n elements. However, I can't go the other way. That is, assuming H has at most n elements, I haven't been able to say anything...

I didn't see a topology forum, so I thought I'd post this question here. Can anyone give any pointers on using van Kampen's theorem? I understand the basic way it works, decompose a space X into open, path-connected sets, say U and V. Then pi1(U) * pi1(V) = pi1(X)/N, where N is a normal...

There are several substanecs different only in a chemical group linked to the same opsition for all those substances. how can i know which one reacts fastest ? ie , OH-R, ON_2-R, O_3P-R, O_4S-R... Thanks

Hi, This time around I need to prove that a finite group of order 10 must contain an element of order 2 and an element of order 5. If the group is cyclic then this is trivial. So assuming the group is not cyclic, it's easy to show that there exists an element of order 2 in the group. And...

I was given a problem to prove there are at most 3 groups of order 21, with extra credit for proving there are at most 2. I am pretty stuck on this one but here is what I have so far: Suppose G is a group of order 21 Let K be a sylow 3-subgroup of G and let H be a sylow 7-subgroup of G...

I need a crash course in surgery obstruction groups. Where should I go to find information on this topic, or does anyone know something about it? Thanks.

if a group of order 2p ( p prime) is abelian...then does it have exactly one element of order 2 ?? if a group is non abelian...i could figure out that there are p elements of order 2. but the abelian case is a bit confusing... also..is it like...any group of order 2p has an element of order p...

Find:2 Non-isomorphic groups of order n squared i think that Zn X Zn is one. Can you help me find another. Thanks

This is my (limited) understanding of particle physics: In particle physics gauge symmetries play an important role. To allow for massive gauge bosons this symmetry is broken. The theory of weak interactions can be derived from a local SU(2) symmetry, and quantumchromodynamics from a local SU(3)...

Letting F be a finite field, how would one show that the multiplicative group must be cyclic? I know that if the order of F = n, then the multiplicative group (say, F*) has order n - 1 = m. Then g^m = 1 for all g belonging to F*. Thanks for your time and help. dogma

For a finite group G, I need to prove that the order of an element in G is a divisor of the order of the group. I'm not sure what this exactly means, but I think you have to use cyclic groups such that G={a^0,a^1,...,a^n) where n+1 is the order of the group and a^0 is the identity element. So I...

The NO2 group directs meta with-deactivation in electrophilic aromatic substitution. The nitroso group - NO directs ortho-para with - deactivation. Write out the electroinc structures of - NO2 and -NO and explain the differences in behavior. Show all pertinent resonance forms for the addition of...

Hi I have this question on homotopy groups: Spacial infinity in two dimensional space is a unit circle S1 (topologically). I understand that. Now, in physics one can prove that fields will exhibit an equation (expressed by the map phy --> v =constant) that also represents a unit circle. Now...

Hello all. I'm looking for an equation, one that I use to know how to figure out(but alas I am getting old and senile), that will alow me to determine how many groups of a given number are in a entire set. For example; in a set of 5, how many possible groups of 3 would there be? The answer is...

This is about amines: The functional groups present are: C=O and NH2 Why does the N get protonated always but not C=O, since the oxygen has more lone pairs and more electronegative than N so shouldn't the oxygen be protonated more easily? I can't think of any good reasons... please help...

Tensors & Differential Geometry -- What are lie groups? I've heard a lot about "lie groups" on this section of the forum, and was wondering what they are and if someone could explain it in simple terms. Thank you.

Hi all, I wanted to study Lie groups and their connections with differential geometry. But i don't want to get involved with lots of 'deep physics'. I am familiar with a little bit of group theory. can somebody suggest the right introductory material like tutorial papers or books for such a...

Lets say for example a hydrocarbon skeleton has two diff functional groups branched to the skeleton.. Let's say one of the functional groups pH is less than 9 to 10, and the other functional group's pH is greater than 2 to 4 that means that molecule to be electrically charged, the pH of the...

Prove: a group of even order must have an even number of elements of order 2

Groups, Rings, Fields? I know what groups, rings, and fields are. My question is why are groups, fields, and rings defined the way they are? Why did mathematicians chose the properties that they did that define groups, rings, and fields? What is so special about those properties? Why...

i can't grasp these concepts, 1-to-1 and onto have always annoyed me. here's 1 question, (i don't know how to post symbols so Beta ..) (C is Complex numbers) Let Beta:<C,+> -> <C,+> by Beta(a+bi)=a-bi (that is, the image is a +(-b)i). Prove Beta is an isomorphism of...

Is the Galois group of F=Q(sqrt2,3i) the maps {id, tau , sigma, gamma}, where (1) id is the identity (2) tau maps sqrt2 to -sqrt2 and leave 3i alone (3) sigma leaves sqrt2 alone and maps 3i to -3i (4) gamma maps sqrt2 to -sqrt2 and 3i to -3i ? If so, the what are the fixed fields of the...

what is the number of elements of order 5 in the permutaion group S7?? so what we're concerned with here is, after decompositon into disjoint cycles the l.c.m of the lengths must be 5. since 5 is a prime, the only possible way we could get 5 as l.c.m would be to fix ANY 2 elements amongst the 7...

Blood groups and transfusions... I have learned that when you transfuse the wrong blood to a recipient whose antibodies work against the donated blood that agglutination occurs. eg - if you transfuse A blood to a B person, since the B person will possesses anti-A antibodies which will cause...

I've got one of those "research in other resources" questions in my chemistry homework. (meaning the info isn't in the lecture or the text, I guess ). It goes like this: Consider Group 1A and Group 1B of the periodic table. The text states that although A groups have very regular patterns...

Hi I have a few questions in organic chemistry. I would be very grateful if someone can answer them: 1. (cf Morrison & Boyd 6th ed page 523). "Strongly activating groups generally win out over deactivating or weakly activating groups." When the bromination of 3-hydroxybenzaldehyde...