Groups Definition and 867 Threads

Hello! If anybody has a minute, I'd appreciate a quick look-through of my proof that a finite abelian group can be decomposed into a direct product of cyclic subgroups. I'm new to formal writing (as well as Latex) and all feedback is greatly appreciated! Thanks in advance for your time...

Hi, I'm a student of Nuclear Engineering (MS level) at University of Dhaka, Bangladesh. I completed my Honours and Master Degree with Mathematics. I have chosen to complete a thesis paper on "Application of Lie groups & Lie Algebras in Nuclear & Particle Physics." I need some guideline...

Hi! I was wondering why it is possible to write any proper orthochronous Lorentz transformation as an exponential of an element of its Lie-Algebra, i.e., \Lambda = \exp(X), where \Lambda \in SO^{+}(1,3) and X is an element of the Lie Algebra. I know that in case for compact...

Homework Statement 1)Let p,q be primes. Show that the only group homomorphism $$\phi: C_p \mapsto C_q$$ is the trivial one (i.e ## \phi (g) = e = e_H\,\forall\,g##) 2)Consider the function $$det: GL(n,k) \mapsto k^*.$$ Show that it is a group homomorphism and identify the kernel and...

Dear experts, I'm not familiar with crystal structure theory. I'm seek expertise to figure out space groups in 2 dimensions Bravais lattice of the attached structures. In the figure, red and greens dots represent different atoms. I'll greatly appreciate your help. Struture 1...

I always see problems like "how many structurally distinct abelian groups of order (some large number) are there? I understand how we apply the theorem which tells us that every finite abelian group of order n is isomorphic to the direct sum of cyclic groups. We find this by looking at the...

Author: Brian Hall Title: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction Amazon link https://www.amazon.com/dp/1441923136/?tag=pfamazon01-20 Level: Grad Table of Contents: General Theory Matrix Lie Groups Definition of a Matrix Lie Group Counterexamples...

Homework Statement Describe all the subgroups of Z/9Z. How many are there? Describe all the subgroups of Z/3ZxZ/3Z. How many are there? The Attempt at a Solution I don't even know where to start with this question. If someone could just point me in the right direction that would be...

Prove the Generalized Associative Law for Groups (i.e. a finite sum of elements can be bracketed in any way). The proof is outlined in D & F. I just want to know whether or not one part of my proof is correct. Show that for any group G under the operation °, and elements a1,...,an, any...

Hello! I am currently trying to get things straight about Lie group from two different perspectives. I have encountered Lie groups before in math and QM, but now I´m reading GR where we are talking about coordinate and non-coordinate bases and it seems that we should be able to find commuting...

I was thinking about the following proposition that I think should be true, but I can't pove: Suppose that F is a group freely generated by a set U and that F is also generated by a set V with |U| = |V|. Then F is also freely generated by V. This is something that I intuitively think must...

Hoping to get some assistance here on a volunteer project I am working on. I am writing a program for my bicyle club in preparation for our spring training series. We will have x participants that will be divided weekly into y number of (approximately) equal groups containing z members per...

Homework Statement What exactly does G={ f: R -> R : f(x)=ax+b, where a is not equal to zero} is a group under composition, mean? So what are the elements of G? Are they (for example) f(x)=ax+b and g(x)=a'x+b'? Or are they f(x)=ax+b and f(y)=ay+b? Thanks in advance Homework Equations...

Hi All, I have a hard time answering the following. I need some help. Let Z={a,b,c,d,e,f} and let X denote the set of 10 partitions of Z into two sets of three. Label the members of X as follows: 0 abc|def 1 abd|cef 2 abe|cdf 3 abf|cde 4 acd|bef 5 ace|bdf 6 acf|bde 7 ade|bcf 8...

G, H, K are groups. G is finite. GxH is isomorphic to GxK. Prove H is isomorphic to K. Give an example to show that this does not hold when G is infinite. The counter example when G is infinite is Rx{0} and RxR (R - real numbers) I'm having trouble Proving the main part of the question. I...

So Munkres, on page 424 of Topology (2nd edition) says that "...two free groups are isomorphic if and only if their systems of free generators have the same cardinality (We have proved these facts in the case of finite cardinality)." Nowhere explicitly does he say this, although it seems that...

So this is a pretty dumb question, but I'm just trying to understand homomorphisms of infinite cyclic groups. I understand intuitively why if we define the homomorphism p(a)=b, then this defines a unique homorphism. My question is why is it necessarily well-defined? I think I'm confused...

I am not quite sure if I am using the correct formula. The problem is -A class of 30 students(12 male and 18 female) are put into groups of 3. How many combinations can be formed if the requirement is that no group can be entirely male or female? I get 4060 since it doesn't matter the order...

Let $A,B,C$ be finite abelian groups. Assume that $A\times B\cong A\times C$. Show that $B\cong C$. I observed that $(A\times B)/(A\times\{e\})\cong B$ and $(A\times C)/(A\times\{e\})\cong C$. So I need to show that $(A\times B)/(A\times\{e\})\cong (A\times C)/(A\times\{e\})$. Let...

Homework Statement Suppose G and F are groups and GxF is isomorphic to G'xF', if G is isomorphic to G', can we conclude that F is isomorphic to F'? Homework Equations The Attempt at a Solution I'm trying to give a proof using the first isomorphism theorem (using that GxF/Gx(e) is isomorphic to...

I am doing an introductory analysis and groups course next semester and I have a couple of questions about books. The course textbook is 'An introduction to Analysis' by W R Wade. Can anyone tell me if/when a new edition is expected and if not, what the current edition of the book is? I tried...

wrtie down the possible isomorphism types of abelian groups of orders 74 and 800 then for 74=2*37 then Z(74) is isomorphism to Z2 * Z37 (by chinese remainder theorem) then for 74 , 2 we have Z74 and Z2*Z37 (i am not sure it is right or wrong then for 800 i know i should apply the fundamental...

My sum total of knowledge of composition series is: the definition, the jordan holder theorem and the fact that the product of the indices must equal the order of the group. With this in mind, can someone help with me with finding a composition series for the following:(1) Z60 (2) D12...

Homework Statement Given the structure of Alantolactone, find two functional groups. 2. The attempt at a solution This was a question that was on my exam recently. I answered Ester and Ether, however Ether was marked incorrect. Instead, only the answers Ester and Alkene were accepted. How is...

Find all the abelian groups of order $2100.$ For each group, give an example of an element of order $210.$ $2100 = 2^2 \cdot 3 \cdot 5^2 \cdot 7,$ then $G_1 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb Z_3 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_7 \cong \mathbb Z_{10}...

Z is the set of integers. Prove that (3/7)Z + (11/2)Z = (1/14)Z Attempt: By definition, (3/7)Z+(11/2)Z={3k/7 + 11m/2 : k,m € Z} = {(6k + 77m)/14 : k,m € Z}. Showing that 3/7Z+11/2Z is a subset of 1/14 Z is easy but I can't prove the converse. Can't show that whatever n€1/14Z I take...

Homework Statement I have attached the problem below. Homework Equations The Attempt at a Solution I have tried to use the natural epimorphism from G x G x ... x G to (G x G x ... x G)/(K1 x K2 x ... x Kn), but I do not believe that this is an injective function. Then I tried...

Hello, it is known that the symmetry groups on the 2d Euclidean plane are given by the point-groups (n-fold and dihedral symmetries) and the wallpaper groups. However we can create more symmetries on the plane than just those. For example we can stereographically project the 2d plane onto...

I have a question. If I have a group G of order p where p is prime, I know from the *fundamental theorem of finite abelian groups* that G is isomorphic to Zp (since p is the unique prime factorization of p, and I know this because G is finite order) also I know G is isomorphic to Cp (the pth...

Homework Statement Show that there exists and embedding or show that an embedding can't exist between Z3 and Z. The Attempt at a Solution I've tried to find an embedding and can't so I've decided that an embedding can't exist but how does one show this? Any suggestions would be great.

If one wants to show that two groups are isomorphic is simply finding a single isomorphism between them sufficient? For example. If G is an infinite cyclic group with generator g show that G is isomorphic to Z. So suppose f(g)=ord(g) then f is bijective and a homomorphism I believe?

I am just into reviewing abstract algebra and came across a theorem I'd forgotten: http://en.wikipedia.org/wiki/Finitely-generated_abelian_group#Primary_decomposition (I linked to the theorem instead of writing it here just because I'm not sure how to write all those symbols here) Anyway...

Hi Everyone, Homework Statement If we are asked the number of ways 2n people can be divided into 2 groups of n members, can I first calculate the number of groups of n members that can be formed from 2n people and then calculate number of ways 2 groups can be selected from the number of groups...

Homework Statement Let \mathbb{R}*=\mathbb{R}\{0} with multiplication operation. Show that \mathbb{R}*=\mathbb{I}2 ⊕ \mathbb{R}, where the group operation in \mathbb{R} is addition.Homework Equations Let {A1,...,An}\subseteqA such that for all a\inA there exists a unique sequence {ak} such that...

I'm currently a third year undergraduate doing semiconductor research for about one semester and a summer and I absolutely hate it! My professor doesn't have that many grad students and his lab is severely under funded. I don't have my own mentor/grad student and I've been blindly doing a...

I know that gauging a lie-goup with a kinetic term of the form: \begin{equation} \Tr{F^{\mu \nu} F_{\mu \nu} } \end{equation} Is not allowed for a non-compact lie group because it does not lead to a positive definite Hamiltonian. I was wondering if anyone knew of a general way to gauge...

Let p be a prime. Let H_{i}, i=1,...,n be normal subgroups of a finite group G. I want to prove the following: If G/H_{i}, i=1,...,n are abelian groups of exponent dividing p-1, then G/N is abelian group of exponent dividing p-1 where N=\bigcap H_{i} ,i=1,...,n. Proof: Since G/H_{i}...

Hello, I have a doubt on the definition of Lie groups that I would like to clarify. Let's have the set of functions G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}, that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the...

Hi there, I'm trying to figure out this question: Let A=[aij] be a 3x3 matrix with integer entries and let B=[bij] be it’s transpose. Let P and Q be the Abelian groups represented by A and B respectively. Show that P and Q are isomorphic by comparing the effects of row and column operations...

Hello, Some people on PF are currently self-studying calculus and topology. So we thought we might make a post here so that interested people could join us. We are doing the following books: Book of Proof by Hammack (freely available on http://www.people.vcu.edu/~rhammack/BookOfProof/)...

I am doing a survey of questions grouped into categories. Each question has a weight applied to it. I want to then total and average each category. Lastly, I want to total and average all the categories together. Here's the challenge: I want all of categories and the total average to have the...

My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B). Now however, the author I'm reading defines the group product multiplication as: (a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B. Does this...

I'm curious as to what all the science lovers on physics forums like to listen to. Feel free to throw in whatever styles or genres you like as well.

I know that this is a lot, but I would love some help. My trouble is at the end of Part II. Theorem (Cauchy’s Theorem): Let G be a finite group, and let p be a prime divisor of the order of G, then G has element of order p. Proof: Suppose G is a finite group. Let the order of G be k. Let p...

In alkenes, the more alkyl groups you're attached to means you are more energetically stable, and that we know the reactivity increases as the stability of the intermediate carbocation increases (with tertiary the most stable) I'm puzzled by this relationship, how is it that when it is more...

Proposition: If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order. Proof: Let G be a group with normal subgroup H. Suppose that every element of G/H has finite order and that every element of H has finite order. We wish to...

I was just wondering if anyone has ever tried this? Anyway, Over the summer, I'm taking calculus 2, multi variable calculus and university physics. This fall I will be in modern physics, university physics 2, diff equations and chemistry. I was just wondering if anyone would like to...

How can you tell if two Lie groups are isomorphic to each other? If you have a set of generators, Ti, then you can perform a linear transformation: T'i=aijTj and these new generators T' will have different structure constants than T. Isn't it possible to always find a linear...

Proposition: If G= <a> and b ϵ G, then the order of b is a factor of the order of a. Proof: Let G be a group generated by a. That is, G=<a>. Let b ϵ G. Since G is cyclic, the element b can be written as some power of a. That is, b=ak for some integer k. Suppose the order of a is n. Hence...

Hi, I wanted to see what people think about my current viewpoint on recognizing structures in abstract algebra. You count the number of sets, and the number of operations for each set. You can also think about action by scalar or basis vectors. So monoids groups and rings have one set...