Groups Definition and 867 Threads

Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties? So for example for a group G with order 15 to show that G \cong C_3 \times C_5 would I just have to define all the possible transformations to define the isomorphism...

Homework Statement Let G be a group with identity e, and suppose that a, b, c, x in G. Determine x, given that x2a=bxc-1 and acx = xac. Homework Equations The Attempt at a Solution I know the three axioms for group. G1. Associativity. For all a, b, c in G, (a * b) * c = a...

If we have two groups of photons; each group consisting of two entangled photons. We allow one of the photons from each group to interact with another object. If we perform a polarisation measurement on each photon of one group, will the photons in the other group, independent of whether...

Homework Statement We are supposed to say how many limit points the set A={sin(n)} where n is a positive integer. My teacher said to use a theorem by Kronecker to help with it. His theorem says from wiki, that an infinite cyclic subgroup of the unit circle group is a dense subset...

I am reading Dummit and Foote Ch 18, trying to understand the basics of Representation Theory. I need help with clarifying Example 3 on page 844 in the particular case of S_3 . (see the attahment and see page 844 - example 3) Giving the case for S_3 in the example we have the...

The problem is kind of easy to understand. Given is some points, say 10 points. (I am using numbering for understanding) 0 1 2 3 4 5 6 7 8 9 Now group these such that the group size is 5 and there is no overlap so, there can be 2 groups. the groups are (0 1 2 3 4) & (5 6 7 8 9) Now...

Help! For p prime I need to show that C_{p^2} \ncong C_p \times C_p where C_p is the cyclic group of order p. But I've realized I don't actually understand how a group with single elements can be isomorphic to a group with ordered pairs! Any hints to get me started?

Regarding finite cyclic groups, if a group G, has generator g, then every element h \in G can be written as h = g^k for some k. But surely every element in G is a generator as for any k , (g^k)^n eventually equals all the elements of G as n in takes each integer in turn. Thanks...

HI, I was reading an article and it says that a finite group of order p^aq^b, where p, q are primes, is solvable and therefore not simple. But I can't quite understand why this is so. I do recall a theorem called Burnside's theorem which says that a group of such order is solvable. But then I...

Homework Statement Let G be a cyclic group of order n and let k be an integer relatively prime to n. Prove that the map x\mapsto x^k is sujective. Homework Equations The Attempt at a Solution I am trying to prove the contrapositon but I am not sure about one thing: If the map is...

Apparently there is an isomorphism between the additive group (ℝ,+) of real numbers and the multiplicative group (ℝ_{>0},×) of positive real numbers. But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?! Thanks for...

Hi, I have a mathematics/Matlab question. Suppose I have a speaker that serves as a sound source, and two IDENTICAL microphones to the left and right of this speaker. Suppose that each microphone collects data regarding the sound level of the speaker, and that there are over 3,000 data values...

Homework Statement Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) } and N=<(13)(24)> which is a normal subgroup of d4 . List the elements of d4/N . Homework Equations The Attempt at a Solution I computed the left and right cosets to...

how does one evaluate the direct product between a group G with components that are say 2-tuple and a group H with components that are just 1-tuple?

If I were to solve a system of multiple equations in the form αx+βy+ζz=p_{1} Where α,β,ζ are constants x,y,z are variables, and p is a prime, how would I use Galois theory and/or number theory to find the number of solutions if the other equations could all be written in the form...

Hello everyone, When we speak about the SU(2)L group (in electroweak interactions for example), about what group do we talk ? What is the difference with the SU(2) group ? And with the SU(2)R ? Why is the label so important ? I ask this because I see that a Lagrangien can be invariant...

Let G and H be finite groups. The maximal subgroups of GxH are of the form GxM where M is maximal subgroup of H or NxG where N is a maximal subgroup of G. Is this true?

I was thinking about some similarities in the definitions of group and field, and if it would be possible to generalize in some sense, like follows. A field is basically a set F, such that (F,+) is a commutative group with identity 0, and (F-{0}, .) is a commutative group with identity 1, and...

Hello I have to show that the free product of a collection of more than one non-trivial group is non-abelian. But doesn't this just follow from the definition of the free product? Or how would you tackle this question?

Homework Statement Let G be a group with identity e. Let a and b be elements of G with a≠e, b≠e, (a^5)=e, and (aba^-1)=b^2. If b≠e, find the order of b. Homework Equations Maybe the statement if |a|=n and (a^m)=e, then n|m. Other ways of writing (aba^-1)=b^2: ab=(b^2)a...

I am in the fourth year of an MPhys and feel a PhD is the best way to further myself. I want to apply to groups that specialize in the theory and implementation of computational modelling. I would prefer this to be a group with a wide range across several branches of Physics as opposed to a...

WTS is that \mathbb R^*/N \ \cong \ \mathbb R^{**} where N = (-1, 1) then prove that \mathbb R^*/\mathbb R^{**} \ is \ \cong \ to \mathbb Z/2\mathbb Z So the best answer in my opinion is to construct a surjection and use the first iso thm. f:\mathbb R^*\rightarrow\mathbb R^{**}...

what does it mean. I'm thinking list all groups of such order for instance. 115 = 5* 23 hence Z5⊕ Z23 ≈ Z115 ?

If |Aut H| = 1 then how can I show H is Abelian? I've shown a mapping is an element of Aut H previously but didn't think that would help. I have been looking through properties and theorems linked to Abelian groups but so far have had no luck finding anything that would help. The closest I...

Are all groups nonempty? If so, is it because all groups have an identity (element)?

I'm wondering if anyone can help me with learning how to write groups as an external direct product of cyclic groups. The example I'm looking at is for the subset {1, -1, i, -i} of complex numbers which is a group under complex multiplication. How do I express it as an external direct...

Homework Statement G is isomorphic to H. Prove that if G has a subgroup of order n, H has a subgroup of order n. Homework Equations G is isomorphic to H means there is an operation preserving bijection from G to H. The Attempt at a Solution I don't know if this is the right...

Homework Statement Show that the product of two infinite cyclic groups is not an infinite cyclic? Homework Equations Prop 2.11.4: Let H and K be subgroups of a group G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. then f is an isomorphism iff H intersect K is...

I am currently looking at what research groups to apply to for starting a PhD next year, is there anywhere particularly good that I am missing out? I did my Masters in Theoretical Physics but I am actually leaning more towards a mixture of both experimental and theory at this stage, I did do...

This is not a homework question, just a question that popped into my head over the weekend. My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set...

Factor groups! Please I just want to ask about factor groups.. how could a factor group G/A acts on A by conjugation, knowing that A is a normal & abelian subgroup of G.. and what do we mean when we say that an element in a group has jus 2 conjugates?? thanks in advance :)

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 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...