Groups Definition and 906 Threads

1. Problem: Suppose a is a group element such that |a^28| = 10 and |a^22| = 20. Determine |a|. I was doing some practice problems for my exam next week and I could not figure this out. (This is my first post on PF btw) 2. Homework Equations : Let a be element of order n in group and let k...

I'm going through a abstract algebra book I found and am trying to learn more about group theory by going through some of the proofs and practice sets, but am having trouble with the following: Prove that G={a+b*sqrt(2) | a,b E R; a,b not both 0} is a group under ordinary multiplication...

Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...

Abstract Algebra Proof: Groups... A few classmates and I need help with some proofs. Our test is in a few days, and we can't seem to figure out these proofs. Problem 1: Show that if G is a finite group, then every element of G is of finite order. Problem 2: Show that Q+ under...

Homework Statement Let G be a group. Show that G/Z(G) \cong Inn(G) The Attempt at a Solution G/Z(G) = gnZ(G) for some g ε G and for any n ε N choose some g-1 such that g(g-1h) = g(hg-1) and the same can be done switching the g and g-1 This doesn't feel right at all...

Can anyone think of an example of an infinite group that has elements with a finite order?

Homework Statement Find at least 7 pairwise non-isomorphic groups of order 16, and prove that no two among them are isomorphic. Homework Equations I found 7 nonisomorphic groups, but I just am having trouble how to precisely prove they are not isomorphic... The Attempt at a Solution...

Homework Statement Show that if n>1 and F is an arbitrary field, the general linear group defined by n and F is non-abelian Homework Equations A general linear group is the group of invertible matrices with entries from F A non abelian group is a group where the binary operation isn't...

Hi everybody! Ok, so from a few days I've begun a group theory class, and i have to say i love the subject. In particular i happened to like Lie groups, but there are things that are not cristal clear to me, hope you'll help to figure'em out!First of all, Lie groups are continuous group, so...

Show that there are two abelian groups of order 108 that have exactly one subgroup of order 3. 108 = 2^ 2 X 3 ^ 3 Using the fundamental theorem of finite abelian groups, we have Possible abelian groups of order 108 can be : Z108, Z4 + Z27, Z2+Z2+Z27, Z4+Z9+Z3, Z2+Z2+Z9+Z3, Z4+Z3+Z3+Z3...

I'm searching for research groups of quantum loop gravity, Does anyone can help me to find at least one of them?. I want to do my PhD on that topic and I have found kind of hard to find those groups and I am looking for a good supervisor, any reference would be very very helpful. Thanks...

Homework Statement A= \left( \begin{matrix} i & 0 \\ 0 &-i \end{matrix} \right) , B= \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right) \\ Show that \langle A, B \rangle is subgroup of GL_2(\mathbb{C}). And Show that \langle A, B \rangle generated by A and B, and order of...

Hey guys, would you say study groups benefit everyone? I've always studied by myself and so far I'm an A+ student. Are study groups something that help everyone, or are they mostly aimed at certain mindsets? For example, I could see the "slowest" persons of the group getting a lot out of study...

Homework Statement The problem states: "Draw the chemical structure of ATP at a pH of 7. Homework Equations The Attempt at a Solution The textbook diagrams the phophate groups as unprotonated, but since H3PO4 has a pKa of <7, I was thinking that maybe each phosphate group would have lost one...

G is a group and for all elements a,b in G, (ab)^i = (a^i)(b^i) holds for 3 consecutive positive integers. Show that G is abelian. I know how to prove that if (ab)^2 = (a^2)(b^2) then G is abelian. I was thinking that you could reduce the given equality integer by integer till 2 or...

Question 1: $$0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0$$ is an exact short sequence,in order to prove $$\cdots \to H^q (A)\mathop \to \limits^{f^* } H^q (B)\mathop \to \limits^{g^* } H^q (C)\mathop \to \limits^{d^* } H^{q + 1} (A) \to \cdots$$ is an exact long...

I have studied a fair portion of groups, but couldn't imagine what they are all about. Please help me in this regards.

I have to find the number of elements in Aut(Z720) with order 6. Please suggest how to go about it. 1) Aut(Z720) isomorphic to U(720) (multiplicative group of units). 2 ) I am using the fundamental theorem of abelian group that a finite abelian group is isomorphic to the direct products of...

Sylow's theorem tells us that there is one 7-Sylow subgroup and either one of seven 3-Sylow subgroups. Call these subgroups H and K respectively. Sylow's theorem also tells that H is normal in G. I'm not going to write it all out as I don't think it's necessary but in the case when we have...

Homework Statement For any positive integern, let U(n) be the group of all positive integers less than n and relatively prime to n, under multiplication modulo n. Show the the Groups U(5) and u(10) are isomorphic Homework Equations The Attempt at a Solution any 2 cyclic groups of...

Homework Statement Let U(n) be the set of all positive integers less than n and relatively prime to n. Prove that U(n) is agroup under multiplication modulo n Homework Equations The Attempt at a Solution n | a-a' implies n | b(a-a') =ab-a'b and n |b-b' implies n |a'(b-b')...

Homework Statement If G is a finite abelian group and p is a prime such that p^n divides order of G, then prove that G has a subgroup of order p^n Homework Equations Theorem of Finite Abelian Groups: Every finite abelian group G is a direct sum of cyclic groups, each of prime power...

To prove that : f : U_{s} (st) \rightarrow U(t) is an onto map. Note that Us(st)= {x \in U(st): x= 1 (mod s)} Let x \inU(t) then (x, t)=1 and 1<x< t How to proceed beyond point ?

I'm trying to understand why a Lie group always has a non-vanishing vector field. I know that one can somehow generate one by taking a vector from the Lie algebra and "moving it around" using the group operations as a mapping, but the nature of this map eludes me.

For discrete groups, we can easily find the decomposition of the direct product of irreducible representations with the help of the character table. All we need to do is multiply the characters of the irreducible representations to get the characters of the direct product representation and then...

Homework Statement My question is about the rules behind the method of finding the solution not necessarily the method itself. (I am prepping for the GRE subject) How many mutually nonisomorphic Abelian groups of order 50? The Attempt at a Solution so, if understand this...

Hi, All: Please forgive my ignorance here: let G be an infinite group, and let H be a subgroup of G of finite index . Does H necessarily have torsion? I can see if , e.g., G was Abelian with G=Z^n (+) Z/m , then , say, would have subgroups of finite index, but I can't tell if...

Homework Statement Let p, q be distinct primes s.t. p \equiv 1 (mod q). Prove that there exists a non-Abelian group of order pq and calculate the character table. Homework Equations Semi-Direct Product: Let H = < Y | S > and N = < X | R > be groups and let \phi : H \rightarrow Aut...

Homework Statement Just want to make something clear. Are all cyclic groups that have the same number of elements isomorphic to each other. The Attempt at a Solution I think yes because theirs is a one-to-one correspondence and the groups are cyclic which means they have generators.

Hi all, Sorry, I'm not quite sure that I've posted this question in the proper place, but I figured field theory matches best with lie groups in this context. Anyway, my question has to do with the relationship between the fundamental forces (electromagnetism, weak, and strong) and their...

Well, apparently, I'm not too clear on a few things. For #4, what is <[8]> in the group Z18? What does the <> mean around the congruence class? Is my work correct? http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110723_161645.jpg?t=1311456158...

Homework Statement Does Euler's totient function tell me how many elements are in my group? And once I know how many elements are in my group. the generators are the ones that are relatively prime with the number of element in my group. Are my statements correct.

Does anyone know of a good book to read about groups for algebra? I've head that a good book was "A Book of Abstract Algebra" by Charles R. Pinter. And I am just learning about groups so it should be basic.

The .pdf can be ignored. Let A + B = (A - B) U (B - A) also known as the symmetric difference. 1. Look for the identity and let e be the identity element A + e = A (A - e) U (e - A) = A Now there are two cases: 1. (A - e) = A This equation can be interpreted as removing from A all elements...

Is there an "easy" method to finding subgroups of finitely generated abelian groups using the First Isomorphism Theorem? I seem to remember something like this but I can't quite get it. For example, the subgroups of G=Z_2\oplus Z are easy...you only have 0\oplus nZ and Z_2\oplus nZ for n\geq...

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Homework Statement Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G. I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.

So say you have a presentation matrix A for a module, and you diagonalise it and you get something like diag[1,5] well you can interpret that as A can be broken down as the direct sum of 1+Z/5Z. That is the trivial group of just the identity plus cyclic 5. What if your module is defined over...

Homework Statement For n>1, show that the subgroup H of S_n (the symmetric group on n-letters) consisting of permutations that fix 1 is isomorphic to S_{n-1} . Prove that there are no proper subgroups of S_n that properly contain H.The Attempt at a Solution The first part is fairly...

Homework Statement Let G be a finite cyclic group and \ell(G) be the composition length of G (that is, the length of a maximal composition series for G). Compute \ell(G) in terms of |G|. Extend this to all finite solvable groups. The Attempt at a Solution Decompose |G| into its prime...

I wanted to make clear just a quick technical thing. If G is a group, N is a normal subgroup, and \phi_g \in \text{Inn}(G), \phi_g(h) = g h g^{-1} then \phi_g is an automorphism of N, right? However, is it the case that we cannot say that \phi_g is an inner automorphism, since we are not...

Homework Statement Let G be a finite abelian group, and assume that |G| is odd. Show that every element of G is a square. The Attempt at a Solution So we want to show that \forall g \in G, \exists h \in G, g = h^2 . Let g \in G be arbitary, and consider the subgroup generated by g, denoted...

Hi, so I didn't see exactly where group theory stuff goes...but since Lie groups are also manifolds, then I guess I can ask this here? If there's a better section, please move it. I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is...

According to wiki: "a non-abelian group, also sometimes called a non-commutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a." I thought in order to be an abelean group, 5 axioms must be satisfied. If one of them is not satisfied it...

I need some material on the properties and relationships between classical groups. I was using Robert Gilmore's "Lie Groups, Lie Algebras and Some of their Aplications", but it barely covers it (section 2.iv). Does someone know about a book or any lecture notes that could be used to...

Homework Statement Let p=2^(k)+1 , in which k is a positive integer, be a prime number. Let G be the group of integers 1, 2, ... , p-1 under multiplication defined modulo p. By first considering the elements 2^1, 2^2, ... , 2^k and then the elements 2^(k+1), 2^(k+2), ... show that the...

Hi there PF What are groups, and what are they used for in physics? For example if you look at QED, http://en.wikipedia.org/wiki/Quantum...cs#Mathematics , it is said here that QED is a abelian gauge theory with symmetry group U(1). What is this symmetry group, and what is it used for when...

Homework Statement Determine how many non-isomorphic (and which) abelian groups there are of order 54. Determine which of these groups the factor group Z6 x Z18 / <(3,0)> is isomorphic to. Homework Equations The Attempt at a Solution Fundamental theorem for abelian groups...

Let A and B be nxn matrices which generate a group under matrix multiplication. Assume A and B are not orthogonal. How can I determine an nxn matrix X such that X-1AX and X-1BX are both orthogonal matrices? Is it possible to do this without any special knowledge of the group in question?

Hi everyone, :smile: I was wondering why they only use 5th and 3rd groups elements to create n-type and p-type semiconductors respectively. Couldn't we mix 6th or 2nd groups' elements instead to make the semiconductors? What would happen if we do this? Perhaps elements of those groups will...