Groups Definition and 867 Threads

Hey Everyone, I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be...

Could a finitely generated group contain a subgroup which is infinitely generated? Why?

What's so special about finite presentations? Does it indicate some properties about the group?

Hi, it's my first time posting in this forum, so I'm sorry if I have done anything against the forum rules and please point it out to me. Currently revising group theory for an exam in a week's time, and these two practise questions I couldn't finish, so if anyone can push me towards the right...

Here is another problem from Lang. Let G be a finite group. N a normal subgroup. We want to ask what structure must G have in order for all the elements of Aut(G) to send N to N. It is assumed that the order of N is relatively prime to the order of G/N. I have worked on this problem for a...

Homework Statement Find a subgroup of Z_4 \oplus Z_2 that is not of the form H \oplus K where H is a subgroup of Z_4 and K is a subgroup of Z_2. The attempt at a solution I'm guessing I need to find an H \oplus K where either H or K is not a subgroup. But this seems impossible. Obviously...

Homework Statement Show that the group {U(7), *} is isomorphic to {Z(6), +} Homework Equations The Attempt at a Solution I drew the tables for each one. I can see that they are the same size and the identity element for U7 is 1, and for Z6 is 0. I don't really see any...

Homework Statement Determine the number of non-isomorphic abelian groups of order 72, and list one group from each isomorphism class. The Attempt at a Solution 72 = 2^3*3^2 3= 1+1+1= 2+1= 3 (3) 2= 1+1= 2 (2) 3*2 = 6 And then I get lost on the...

Hi there, I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is...

Homework Statement How many different nonisomorphic groups of order 30 are there? Homework Equations The previous parts of the problem dealt with proving that 3-Sylow and 5-Sylow subgroups of G were normal in G when o(G)=30, though I'm not sure how that relates... The Attempt at a...

From Gauge Theory of particle physics, Cheng and Li I don't understand the flollowing: "Given any two groups G={g1,..} H= {h1,h2,...} if the g's commute with the h's we can define a direct product group G x H={g_ih_j} with the multplication law: g_kh_l . g_mh_n = g_kh_m . h_lh_n Examples...

I'm a bit confused about what alkyl groups are in organic chemistry. I thought functional groups by definition where groups of atoms which contained at least 1 element other than carbon or hydrogen which were connected to the carbon skeleton of the molecule. What are alkyl groups then? I read...

[SOLVED] finitely generated abelian groups Homework Statement My book states that (\mathbb{Z} \times\mathbb{Z} \times\cdots \times \mathbb{Z})/(d_1\mathbb{Z} \times d_2\mathbb{Z} \times \cdots d_s\mathbb{Z} \times {0} \times \cdots \times {0}) is isomorphic to \mathbb{Z}_{d_1}...

Homework Statement Show that \mathbb{Z}_{p^r}[p] is isomorphic to \mathbb{Z}_p for any r \geq 1 and prime p. \mathbb{Z}_{p^r}[p] is defined as the subgroup \{x \in \mathbb{Z}_{p^r} | px = 0 \}Homework Equations The Attempt at a Solution I don't think I should need to use Sylow's Theorems for...

I'm taking a course on Lie Groups and the Representations. We are using the book: Representations of compact Lie Groups by Bröcker and Dieck, and I find it very unorganized and sometimes sloppy. Can anybody recommend a very clear and rigorous book, where it is not prove by example, "it is easily...

Hi next one, bit confused with this problem: any hints on any of the parts would be greatly appreciated. QUESTION: --------------------------------------- let G be a group of order 8 and suppose that y \epsilon G has ord(y)=4. Put H = [1,y,y^2,y^3] and let x \epsilon G-H (i) show that H...

Let G be a group and let N\trianglelefteq G , M\trianglelefteq G be such that N \le M. I would like to know if, in general, we can identify G/M with a subgroup of G/N. Of course the obvious way to proceed is to look for a homomorphism from G to G/N whose kernel is M, but I can't think of...

Homework Statement Prove: a topological group is discrete if the singleton containing the identity is an open set. The statement is in here http://en.wikipedia.org/wiki/Discrete_group The Attempt at a Solution Is that because if you multiply the identity with any element in the group, you get...

http://img180.imageshack.us/img180/9589/simplell9.jpg Is 1. c) as simple as i think it is? I have gone through my notes and can't find anything to do with it, the module for it is Numbers, symmetries and groups, any ideas or do i simple just wack in 13/7 on my calculator and write down...

[SOLVED] extension field Homework Statement Let E be an extension field of Z_2 and \alpha in E be algebraic of degree 3 over Z_2. Classify the groups <Z_2(\alpha),+> and <Z_2(\alpha)^*,\cdot> according to the fundamental theorem of finitely generated abelian groups. Z_2(\alpha)^* denotes the...

Homework Statement Let G and H be groups. We define a binary operation on the cartesian product G x H by: (a,b)*(a',b') := (a*a', b*b') (for a,a' \inG and b,b'\in)H Show that G x H together with this operation is a group. Homework Equations The Attempt at a Solution To...

what are electron withdrawing groups? please give any important advice to solve questions based on it concept.

Let p,q be distinct primes with q < p and let G be a finite group with |G| = pq. (i) Use sylow's theorem to show that G has a normal subgroup K with K \cong G (ii) Use the Recogition Criterion to show G \cong C_p \rtimes_h C_q for some homomorphism h:C_q \rightarrow Aut(C_p) (iii)...

Homework Statement Let H and K be normal subgroups of a group G. Give an example showing that we may have H isomorphic to K while G/H is not isomorphic to G/K. Homework Equations The Attempt at a Solution I don't want to look in the back of my book just yet. Can someone give me a...

[SOLVED] simple groups Homework Statement T or F: All nontrivial finite simple groups have prime order. Homework Equations The Attempt at a Solution I want to say yes with Lagrange's Theorem, but I am not sure that applies...

Homework Statement Exercise 1.2:2. (i) If G is a group Define an operation dG on |G| by dG(x, y) = x*y^-1. Does the group given by (G,dG) determine the original group G with * (I.e., if G1 and G2 yield the same pair, (G1,dG1) = (G2,dG2) , must G1 = G2?) There is a part II, but I would...

Homework Statement Prove that the following sets form infinite groups with respect to ordinary multiplication. a){2^k} where k E Z b){(1+2m)/(1+2n)} where m,n E Z Homework Equations The Attempt at a Solution I sort of know about closure associativity identity inverses...

Let \rho : \mathbb{H} \to \mathbb{H}; q \mapsto u^{-1}q u where u is any unit quaternion. Then \rho is a continuous automorphism of H. I'm asked to show that \rho preserves the inner product and cross product on the subspace \mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} +...

prove that if G is a finite and abelian group and m is the least common multiple of the order of it's element, that there is an element of order m. My idea: if ai are the elements of G, the order of a1*a2 is lcm(a1,a2) and the result follows directly when applied to all ai... but why is this...

Homework Statement What is the point of giving free abelian groups a special name if they are all isomorphic to Z times Z times Z ... times Z for r factors of Z, where r is the rank of the basis? Homework Equations The Attempt at a Solution

Hehe, I'm working through the complete groups books right now, so don't think I ask you all my homework questions... I'm doing a lot myself too =). Homework Statement 1) If H is a subgroup of S_n, and is not contained in A_n, show that precisely half of the elements in H are even permutations...

ok I've managed to solve the other 2 questions. here is my final one: (1) If G is a group and n \geq 1 , define G(n) = { x E G: ord(x) = n} (2) If G \cong H show that, for all n \geq 1 , |G(n)| = |H(n)|. (3) Deduce that, C_3 X C_3 is not \cong C_9. Is it true that C_3 X C_5...

Homework Statement Expain why every normal subgroup is the kernel of some homomorphism. The Attempt at a Solution Every kernel is a normal subgroup but the reverse I can't show rigorously. It seems possible how to show?

Hi i have completed the answer to this question. Just need your verification on whether it's completely correct or not: Question: If G is a group and xEG we define the order ord(x) by: ord(x) = min{r \geq 1: x^r = 1} If \theta: G --> H is an injective group homomorphism show that, for...

I was wondering about the classification of groups with a certain number of subgroups. I (sort of mostly I think maybe) get the ideas behind classification of groups of a certain (hopefully small) order, but I came across a question about classifying all groups with exactly 4 subgroups, and I...

Why are Lie groups also manifolds?

What Lie groups are also Riemann manifolds? thanks

I'm looking for a solid book on Lie groups and Lie algebras, there is too many choices out there. What is a classic text, if there is one?

Hello, Is there a well known method (algorithm, process, etc.) by which objects are sorted into percentile groups based on some aspect of the object such as weight or size?

What is the connection between the two if any? What kind of algebra would Lie groups be best labeled under?

Just a pregrad-level curiosity: I see often repeated (in the Wikipedia page defining "Field", for one) that, from the field's axioms, it can be deduced that F,+ and F\{0},* are both commutative groups. Yet, the closure property of * is only guaranteed on F, not necessarily on F\{0}. If I'm...

Homework Statement Prove that if H is a subgroup of a finite group G, then the number of right cosets of H in G equals the number of left cosets of H in G Homework Equations Lagrange's theorem: for any finite group G, the order (number of elements) of every subgroup H of G divides...

Hi, I recall being told in an algebra course in college that there exist groups with matching order tables and that are nonetheless not isomorphic. That is, if you list out the orders of all the elements in one group and all the orders of the elements in the other, the lists are "the same"...

Everyone must be familiar with U(1),SU(2) and SU(3) Lie groups in particle physics . But how does one define the multiplication of two groups of different dimensions aka SU(2) X U(1) or SU(3) X SU(2) X U(1).

Hi, everyone: I am asked to show that a group G acts by isometries on a space X. I am not clear about the languange, does anyone know what this means?. Do I need to show that the action preserves distance, i.e, that d(x,y)=d(gx,gy)?. Thanks.

How does soluble groups fit into algebra? Why is there another name for it called solvable groups? What branch does it fall under?

[SOLVED] Why emit alphas, not other nucleon groups? Homework Statement (Advanced Physics; Adams and Allday; Spread 8.18, Question Section 8.18, question 3) Why do you think helium-4 nuclei (alpha particles) are often emitted from unstable heavy nuclei whereas bundles of neutrons or protons...

Hi, I want to take this course next term. One reason is because I think it will help me with mechanics, classical and quantum, which are taken next year at advanced level. The problem is I'm taking calc2 atm, and its a listed prereq for this group course. I got all the other prereq's...

So, the question is: Determine all finite groups that have at most three conjugacy classes I'm a little confused by how to start. Right now, we can say for sure that cyclic groups of order 1, 2, and 3 satisfy this criterion. Also, with Lagrange's Theorem and the counting formula(I'm using...

Ok, well a corollary to Lagrange's theorem is that every group of prime order, call it G, must be cyclic. Consider the cyclic subgroup of G generated by a (a not equal to e), the order of the subgroup must divide the order of p, since the only number less than or equal to p that divides p is p...