# What is Group theory: Definition and 375 Discussions

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

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2. ### What is meant by compex dimension? (Abstract algebra)

picture since the text is a little hard to read i have no problem showing this is a vector space, but what is meant by complex dimention? Is it just the number on independant complex numbers, so n?
3. ### Fixed point free automorphism of order 2

I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
4. ### I Full Course in Group Theory (and More) on YouTube

I created a YouTube channel (here's the link) a few months ago in which I post detailed lectures in higher mathematics. I just finished my Group Theory Course. Here is a sample video. Apart from that, so far I have uploaded A first course on Linear Algebra (which I am currently renovating). A...
5. ### I Pin & Spin Groups: Double Covers of Orthogonal & SO Groups

Pin Groups are the double cover of the Orthogonal Group and Spin Groups are the double cover of the Special Orthogonal Group. Both sets of the double cover are considered to be groups, but it seems that only one of the sets of the double cover actually contains the identity element, which means...
6. ### Ideas for group theory for high school math project

Hi As high school teacher, I sometimes have those extremely talanted and self driven pupils. In their final year, they are required to make a science or math project, roughly one month full-time studies, approx 15-20 pages report. This academic year, one of my students have learned some group...
7. ### Group Theory — Introduction to Higher Mathematics

If you have always wondered what group theory is useful for and why it even exists, this is the video for you. We cover everything from the basic history of group theory, over how and why subgroups partition groups, to the classification of all groups of prime order.
8. ### B One-to-many relations in group theory

I apologize for the simple question, but it has been bothering me. One can write a relationship between groups, such as for example between Spin##(n)## and SO##(n)## as follows: \begin{equation} 1 \rightarrow \{-1,+1 \} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n) \rightarrow 1...
9. ### Studying Should I study Topology or Group Theory?

Hello! I'm a physics graduate who is interested to work in Mathematical Physics. I haven't taken any specialized maths courses in undergrad, and currently I have some time to self-learn. I have finished studying Real Analysis from "Understanding Analysis - Stephen Abbott" and I'm currently...
10. ### Courses Should I take a group theory course before QFT?

I know that studying QFT requires understanding Lie Groups and infinitesimal generators as they correspond to symmetry transformations. I want to study or take a course (offered by my university) in QFT in the coming academic year and I have the option to take a abstract algebra course offered...
11. ### Proper Lorentz transformations from group theory?

Hi, I was looking at this derivation https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates and I was wondering 1- where does the group structure come from? The principle of relativity? or viceversa? or what? 2- why only linear transformations? I remember...
12. ### Direct product of two semi-direct products

After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$ Any help appreciated.
13. ### Good introductory book about Lie Group Theory?

Summary:: Good introductory book about Group Theory? Hi, I am looking for a good introductory book about Group Theory for physicists.
14. ### I Degrees of Freedom of SO(3)

The group ##\rm{O(3)}## is the group of orthogonal ##3 \times 3## matrices with nine elements and dimension three which is constrained by the condition, $$a_{ik}a_{kj} = \delta_{ij}$$ where ##a_{ik}## are elements of the matrix ##\rm{A} \in O(3)##. This condition gives six constraints (can be...
15. T

### I On group theory

Hello there.Questions I have: what is the value of group theory?I am not trying to say that it is not important I want to know what made mathematicians study these objects and we still study them today.I know there are very interesting for me at least examples of groups like the Lie group but...

39. ### A Geometric Group Theory

Hey all I previously asked about some math structure fulfilling some requirements and didn't get much out of it ( Graph or lattice topology discretization ). It was a vague question, granted. Anyway, I seem to have stumbled upon something interesting called geometric group theory. It looks...
40. ### Clebsch-Gordan Decomposition for 6 x 3

Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...
41. ### Other Using Michael Artin's "Algebra" for Group Theory

Hi all, I have stumbled upon Artin's book "Algebra" and was wondering if I could use it to do some self-study on Group Theory. Some background: I am a physics undergraduate who has some competence in elementary logic, proofs and linear algebra. It seemed to me that ideas related to Group...
42. ### A Fields transforming in the adjoint representation?

Hi! I'm doing my master thesis in AdS/CFT and I've read several times that "Fields transforms in the adjoint representation" or "Fields transforms in the fundamental representation". I've had courses in Advanced mathematics (where I studied Group theory) and QFTs, but I don't understand (or...
43. ### Show that ##G\simeq \mathbb{Z}/2p\mathbb{Z}##

Homework Statement Let ##G## be a group of order ##2p## with p a prime and odd number. a) We suppose ##G## as abelian. Show that ##G \simeq \mathbb{Z}/2p\mathbb{Z}## Homework Equations The Attempt at a Solution Intuitively I see why but I would like some suggestion of what trajectory I could...
44. ### Commutator group in the center of a group

Homework Statement [G,G] is the commutator group. Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##. Homework Equations The Attempt at a Solution In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that...
45. ### An exercise with the third isomorphism theorem in group theory

Homework Statement Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##. a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H## b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K## Homework Equations The three...
46. ### A Taxonomy of Theories in Theoretical Physics

It goes without saying that theoretical physics has over the years become overrun with countless distinct - yet sometimes curiously very similar - theories, in some cases even dozens of directly competing theories. Within the foundations things can get far worse once we start to run into...
47. ### Isomorphism of dihedral with a semi-direct product

Homework Statement Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2$$ where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$ Homework Equations I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this...
48. ### Solid State Group theory paper suggestions for my classes

I teach group theory for physicists, and I like to teach it following some papers. In general my students work with condensed matter, so I discuss group theory following these papers:  Group Theory and Normal Modes, American Journal of Physics 36, 529 (1968)  Nonsymmorphic Symmetries and...
49. ### Show injectivity, surjectivity and kernel of groups

Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...
50. ### I How to properly understand finite group theory

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with...