Groups Definition and 906 Threads

Checking to see whether semidirect products are isomorphic. Homework Statement I want to simplify this semidrect product (Z_7 \rtimes_{\bar{\alpha}} Z_3) \rtimes_{\alpha} Z_2, but I'm not sure how. In other words, I want to see if this is isomorphic to (for example) Z_7 \rtimes_{\alpha}...

EDIT: To moderators, I frequent this forum and the Calculus forum and I have accidently put this in Introductory Physics. Can it be moved? Sorry for inconveniences. Homework Statement 1)If ##G \cong H \times \mathbb{Z}_2, ## show that G contains an element a of order 2 with the property that...

Homework Statement Let: G = { \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \in GL(2,ℝ)} H = { \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \in GL(2,ℝ)} K = { \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} \in GL(2,ℝ)} Is G isomorphic to H x K? Homework Equations...

First let me apologise for the improper use of 'groups'. I'm not a mathematician but I know that 'groups' means something specific. Anyway, here is my problem. The exact circumstances of my problem are esoteric and bothersome to explain, and I don't want to distract you with details that do...

Show that a•b=(a+b)^2 has no identity for real numbers Hi this is a new topic please help Thanks

Hi, All: I've been curious about this for a while: Why are there different names for different groups of animals? We have ( I think) schools of fish,packs of wolves, etc. I imagine it may have to see with the fact that groups of animals are organized differently. Still...

Question about matrix groups and conjugate subgroups? This question concerns the group of matrices L = { (a 0) (c d) : a,c,d ∈ R, ad =/ 0} under matrix multiplication, and its subgroups H = { (p 0, (p - q) q) : p,q ∈ R, pq =/ 0} and K = { (1 0, r 1) : r ∈ R}Show that one of H and K is a normal...

There is a corollary in our textbook that states "Let G be a group of order 12 whose 3-Sylow subgroups are not normal. THen G is isomorphic to A_4." I attached the proof of this corollary and an additional corollary and proposition that was used for the proof. The 2nd last paragraph is a bit...

I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism...

Hello, I am unsure of how to utilize protecting groups in organic synthesis problems. What would be an immediate sign to the trained eye that one should use a protecting group such as TBDMS-Cl?

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...

let X(2n) be the group whose presentation is : ( x,y l x^n = y^2 = 1 , xy = yx^2 ) show that if n = 3k then X2n has order 6 and if (3,n)=1 then x satisfies the additional relation x=1 , in this case , deduce that X(2n) has order 2 note that : x^3 = 1 ____________ I tried to...

I am wondering if some one can help it this: Suppose G is a group with 316 \leq|G|\leq 325. Given that G is simple, find the possible value(s) for |G|. Be sure to explain your reasoning for each number. You'll need Sylow's Theorems of course. This is what I have done: the prime factorization...

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...

Homework Statement Let G be a compact connected Lie group define the loop group and the based loop group as LG = \{ \gamma \in C^\infty(S^1,G) \}, \Omega G = \{ \gamma \in LG : \gamma(e_{S^1}) = e_G \} (choose whatever identification of the circle S^1 you like ). Show that \Omega G is a...

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...

Homework Statement Given the dihedral group of symmetries of a square; what is the stabiliser group of a vertex (or edge)? Homework Equations The stabiliser group is G_x={g\inG|gx=x} I guess for a vertex/edge that means the transformations in D4 (generated by reflection in x-axis and...

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...

This one comes from Duderstadt and Hamilton, Problem 7-3. In multi-group diffusion theory What percentage of neutrons slowing down in hydrogen will tend to skip energy groups if the group structure is chosen such that \frac{E_{g-1}}{E_{g}}=100= 1/\alpha_{approx}. I know that the...