Group theory Definition and 380 Threads

What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

Homework Statement If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element. Homework EquationsThe Attempt at a Solution I feel like there is missing information, but that cannot be. This seems too simple: The order of G is the smallest possible integer n...

Hi. I have the following question: Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N? Here is what I think. Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N. Now...

As part of physical chemistry I am reading up group theory for molecular symmetries. I realize the way chemistry textbooks treat this must be very different from what mathematicians do. So I want to know how I take a point group, find the matrix operations and get the character table.For an C2...

Hey, (I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!) I am trying to understand the following statement. "Suppose a single qubit has a state represented by the...

Homework Statement Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##: ##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...

The question: Let f: G -> H be a homomorphism of groups with ker(f) finite, the number of elements being n. Show that the inverse image is either empty or has exactly n elements. My work so far: Let h be eH (identity on H). Then the inverse image is ker(f) so has n elements, which makes it...

This is what I have so far: I'm trying to show that the matrix D has to be unitary. It is the matrix that transforms the wavefunction.

Hi, I have one spot remaining to take a pure math course, and I'm trying to decide between complex analysis and group theory. Although I've touched some of the basic of dealing with complex numbers in my physics/DE courses, they haven't gone in much depth into them beyond applications. On the...

Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...

I hope this post won't become too tedious. I've completed my undergrad studies in physics and if things go well i will begin my master's degree in April. The thing is, since my path to graduation has been peculiar (to say the least) I'm kinda weak in maths skills atm and need to improve. I'm...

Hi, I am looking for textbooks in relativistic quantum mechanics and group theory. I have just finished my undergraduate studies in Physics and am looking to specialise in theoretical high-energy physics. Therefore, textbooks in relativistic quantum mechanics and group theory suited for that...

Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...

Homework Statement Is the following a valid group? The values contained in the set of all real numbers ℝ, under an operation ◊ such that x◊y = x+y-1 Homework Equations Axioms of group theory: Closure Associativity There must exist one identity element 'e' such that ex=x for all x There...

Homework Statement Good day all, Im completely stumped on how to show this: |AN|=(|A||N|/A intersect N|) Here: A and N are subgroups in G and N is a normal subgroup. I denote the order on N by |N| Homework Equations [/B] Second Isomorphism TheoremThe Attempt at a Solution Well, I know...

EDIT: I've just realized this is the 'Calculus and beyond' subforum - I saw 'beyond' and thought, "Well I've done all my calculus, and now I'm doing group theory, so this thread must go here!". But now I realize it surely belongs somewhere else. Sorry about that. Mods feel free to shift it to...

I study physics and currently taking a mathematical physics course. One of the topics is group theory and we will see the following topics: Symmetries, discrete groups, homomorphisms, isomorphisms, continuous groups, and linear transformations in phase space. This topic will be covered with...

I'm doing a small research project on group theory and its applications. The topic I wanted to investigate was Noether's theorem. I've only seen the easy proofs regarding translational symmetry, time symmetry and rotational symmetry (I'll post a link to illustrate what I mean by "the easy...

Let $G$ be a set and $*$ a binary operation on $G$ that satisfies the following properties: (a) $*$ is associative, (b) There is an element $e\in G$ such that $e*a=a$ for all $a\in G$, (c) For every $a\in G$, there is some $b\in G$ such that $b*a=e$. Prove that $(G, *)$ is a group. My...

If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...

I am studying Group Theory at the moment and i am not sure about a theorem. Is it true that a Lie Group G is compact if and only if every finite complex representation of it is unitary? I know that is true the if, but what about the viceversa? Same question. Is it true that a Lie group is...

I am physics student.I know basic definition of topological space.I want a book(or may be any web note or video lecture) where topology spaces of various groups are rigorously discussed.

I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods are applied in physics, such as in the context of the standard model and beyond for example. I am...

An element of SU(2), such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as U(x) = e^{ixT_1} = \left( \begin{array}{cc} \cos\frac{x}{2} & i\sin\frac{x}{2} \\ i\sin\frac{x}{2} & \cos\frac{x}{2} \\ \end{array} \right) = \left(...

Case 2: I get that D = c I means A must also be proportional to I but how does that mean B must be diagonal? Question: Answers:

Hi there. Can anybody recommend a good textbook for an undergraduate wanting to study group theory (especially representation theory). I'm thinking of reading "visual group theory" by Carter for conceptual understanding but I also need a book to study alongside this that gives a more formal...

Hello all. I am new here. I am in the last quarter of a 3 quarter sequence of undergrad quantum mechanics and I just had some conceptual questions (nothing pertaining to homework). We just recently covered Berry's Phase and the Dynamical Phase. Now I wanted to start with a more basic quantum...

Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework EquationsThe Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this...

Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...

Homework Statement Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+) Homework Equations Isomorphism, Group Theory, Binary Operation The Attempt at a Solution Let a,b,c,d∈ℝ Define f : M→ℂ by f( (a -b) (b a) ) = a+bi 1-1: Suppose f( (a -b) (b a) )= f( (c...

Homework Statement Good afternoon, How can you mathematicaly talk about how how group theory compares to electromagnetism. Homework Equations e^iθ=Cosθ+iSinθ The Attempt at a Solution I know that the above formula is because of a sin wave and a cosine wave. Put them together and you get a...

On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...

Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?

Homework Statement Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression: R.RT= I with respect to the parameter t...

I am looking for an introductory book on group theory for my son who is a high school freshman. He has a good grasp of the basics of mathematics and is ready to take calculus classes. He has a very strong intuitive grasp of symmetry and transformations, so I thought that he may be ready to be...

Homework Statement If G is a group of even order, show that it has an element g not equal to the identity such that g^2 = 1. Homework Equations None The Attempt at a Solution What I wrote: If |G| = n, then g^n = 1 for some g in G. Thus, (g^(n/2))(g^(n/2)) = 1, so g^(n/2) is the element of...

Can anybody name some real world applications of group theory? I would be particularly interested to hear any uses of cyclic groups to solve everyday problems one could encounter.

Hello Big Minds, In the following analysis...It is said that D_2 contains three subgroups Z_2...why did he choose a mathematical constuction contains only two of the the three subgroups? shouldn't he use the three in his construction? what will happen if he used the three? [from the book of...

Is there any application of maths in music, which topic is directly used. I've heard group theory is used, but how it is used. Plz help..i m going to work on a project that will show how maths works in music or sound system.

I am working on this problem with lots and lots of nesting definitions like this following, and I have been trying to get help from here as well as http://www.quora.com/How-do-I-prove-G-Z-R-G-is-isomorphic-to-Aut-R-G , but none gave me complete help: Show that ##G/Z(R(G))## is isomorphic to a...

Homework Statement Let G be a non-abelian group with order ##p^3##, p prime. Then show that the order of the center must be p. Homework Equations Theorem in our book says that for any p-group the center is non-trivial and it's order is divisible by p. Class eq. ##|G|= |Z(G)| + \sum{[G ...

Hello, I’m looking for introductory books/notes on group theory and algebra. We are using “Groups” by Jordan and Jordan in class, but I am looking for something a little more in-depth. I wouldn’t mind a book that was entirely focused on the pure maths side but if it had lots of applications to...

Why density matrix renormalization group theory works only for 1D systems?

I have this problem on simple group's homomorphism: Let ##G′## be a group and let ##\phi## be a homomorphism from ##G## to ##G′##. Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G′## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subset N##. And last year somebody...

Hello I'm looking for good books on the group theory of quantum mechanics. I have a BS in Physics, MS in Electrical Engineering and decades of work experience in building lasers, and R&D in laser systems, optics & infrared sensing systems. My main goal is to study & understand quantum...

To make a very long story short, in a group theory problem I am working on, I need to prove this: ##A \lhd B \Rightarrow A'\neq A##, where ##A## and ##B## are finite and ##A'## is called the commutator subgroup: ##\begin{align} A' :&= [A, A] \\ &= \langle [x, y] \mid x, y \in A \rangle \\ &=...

Homework Statement Hi, the problem is, I'm unsure of how this is read. The context is that I'm currently reading on group homomorphisms and they use this notation. I think it is also used in composition of functions, but I'm unsure of how it is read there as well. h:X-->Y So how would I say...

Homework Statement Let p: G-->M be a group homomorphism with ker(p) = K. If a is an element of G, how that Ka = {g in G | p(g) = p(a)} Homework Equations none needed The Attempt at a Solution Okay, I've been struggling with this problem for awhile and I've ran into a problem: -Let g be an...

I am working on a problem on automorphism group of radical of finite group like this one: Here are what I know and what I don't know: ##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the...