Groups Definition and 867 Threads

Attached is my attempt at a proof. Please critque! :shy: Thank you!

I was reading on wikipedia on direct product of groups because I wanted find out if every subgroup of G \times H is realized as a direct product of subgroups of G and H. Apparently it is not, because the diagonal subgroup in G \times G disproves this. I'm a little confused, because I thought...

Show that in a finite cyclic group G of order n, writtten multiplicatively, the equation xm = e has exactly m solutions x in G for each positive integers m that divides n. Attempt... Proof: Let G be a finite cyclic group of order n, and suppose m is a positive integer that divides n. Let x be...

The question states: "Let G be a group and let Gn={gn|g ε G}. Under what hypothesis about G can we show that Gn is a subgroup of G?The set Gn is taking each element of G and raising it to a fixed number. I started my investigation by examining what happens if I take n=3 and considering the...

Hi, I will be writing my bechelors thesis on the application of renormalization groups and p-adic analysis on stock markets. The problem is that I don't understand it yet. I need advice on the best path how to learn it. What I know already: - I've read most of Mac Lanes Algebra, so I know...

Is it true that if there is monomorphism from group A to group B and monomorphism from group B to group A than A and B are isomorphic? i need some explanation. thx

Homework Statement find the functional groups in the following compound: Homework Equations C8H16O4The Attempt at a Solution I know there is an ether but there is also something else. What is it? I have tried to find a group that is in the compound, but I have had no luck. It seems to be only...

I've done some searching and have thus far come up empty handed, so I'm hoping that someone here knows something that I don't. I'm wondering if there has been any work on the enumeration of groups of order n (up to isomorphism); specifically, has anyone derived a generating function? Ideally...

how would you estimate number of groups of order 24? i do not need exact number, it is 15. I know that there are 5 groups of order 8 and there is 1 , 4 or maybe 7 sylow 3-groups. but i do not know what to do next

proove that if G and H are divisible groups and there is monomorphisms from G to H and from H to G than G and H are isomorphic

characterize primes p and q for which each group of order pq is nilpotent

There is a theorem for finite groups of isometries in a plane which says that there is a point in the plane fixed by every element in the group (theorem 6.4.7 in Algebra - M Artin). While the proof itself is fairly simple to understand, there is an unstated belief that this is the only point...

good day! i need to prove that the alternating group An is a normal subgroup of symmetric group, Sn, and i just want to know if my proving is correct. we know that normal subgroup is subgroup where the right and left cosets coincides. but i got this equivalent definition of normal group from...

Homework Statement Let a = (a1a2..ak) and b = (c1c2..ck) be disjoint cycles in Sn. Prove that ab = ba. The Attempt at a Solution Sn consists of the permutations of the elements of T where T = {1,2,3,...,n} so assume we take an i from T. Then either i is in a, i is in b, or i is in...

If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23 end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n. How would I then write this as Cn1 x Cn2...

If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23 end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n. How would I then write this as Cn1 x Cn2...

Hey guys, I'm having an issue with a question, namely Let x be a subset of S4. Is x a group? x = {e, (123), (132), (12)(34)} I don't really understand how I can test the 4 axioms of a group and how x being a subset of S4 would help?

Homework Statement Prove that no group of order 96 is simple. Homework Equations The sylow theorems The Attempt at a Solution 96 = 2^5*3. Using the third Sylow theorem, I know that n_2 = 1 or 3 and n_3 = 1 or 16. I need to show that either n_2 = 1 or n_3 = 1, but I am unsure how to do...

Homework Statement Show that Z18/M isomorphic to Z6 where m is the cyclic subgroup <6> operation is addition The Attempt at a Solution M = <6> , so M = {6, 12, 0} I figured I could show that Z18/M has 6 distinct right cosets if I wanted to do M + 0 = {6, 12, 0} M + 1 = {7, 13, 1}...

hey guys, I just want grasp the whole concept of quotient groups, I understand say, D8/K where K={1,a2} I can see the quotient group pretty clearly without much trouble however I start to get stuck when working with larger groups, say S4 For instance S4/L where L is the...

Hi I am trying to learn about quotient groups to fill the gaps on things I didn't quite understand from undergrad. Anyway I have a question regarding this: Can someone please explain how { 0, 2 }+{ 1, 3 }={ 1, 3 } in Z4/{ 0, 2 }? I would think since 0 + 1 = 1 and 2 + 3 = 1 under mod 4...

How to find the center of groups of order 8?

Consider field extensions of the form Q(u) where Q is the field of rational numbers and u=e^{\frac{2\pi i}{n}}, the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else...

hey, if you have the group S3 and treat it also as a g-set under the action x*g = g-1xg is it correct that the stabilizer of the g-set is just the identity element? thanks

I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups On page 6 (see attachment - pages 5 -6 Grove and Benson) we find the following statement: "It is easy to verify (Exercise 2.1) that the vector x_1 = (cos \ \theta /2, sin \ \theta /2 )...

If two elements in a group operate together and can create more then 1 answer (this answer is still a part of set, not foreign) is it still a group, if so why?

I would like to ask if somebody can verify the solution I wrote up to an exercise in my book. It's kind of long, but I have no one else to check it for me :) Homework Statement If H is a maximal proper subgroup of a finite solvable group G, then [G:H] is a prime power.Homework Equations Lemma...

I have a question regarding terminology here. The assignment is somewhat as follows: "If you think any of the following is a group, classify it along the following lines: finite, infinite discrete, finite-dimensional continuous, infinite-dimensional continuous." The definition of finite is...

I am reading Kane's book on Reflections and Reflection Groups and am having difficulties with some basic geometric notions - see attachment Kane paes 6-7. On page 6 in section 1-1 Reflections and Reflection Groups (see attachment) we read: " We can define reflections either with respect to...

"determine all abelian groups of order 1000. which of them contains exactly 3 elements of order two and which of them contain exactly 124 elements of order five?" ok now I have all the terminology down for this. But its the first time attempting such a type of question. How do I go about this...

I am reading Kane - Reflection Groups and Invariant Theory and need help with two of the properties of reflections stated on page 7 (see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7) On page 6 Kane mentions he is working in \ell dimensional Euclidean space ie...

Homework Statement According to wikipedia, one of the requirements of group is: For all a, b in G, the result of the operation, a • b, is also in G. So say we have 2 (2x2) matricies as elements of a group: \frac{0|1}{1|0} and \frac{w|0}{0|w^{2}} and the product \frac{0|1}{1|0} •...

Homework Statement For each m >= 2, find a group with a composition series of length 1 with a subgroup of length m. Homework Equations Simple groups iff length 1. If G is abelian of order p1^k1...pr^kr, then length G = k1 + ... + kr If G has a composition series and K is normal...

Hello, from Weinberg's Quantum Field Theory book I am confused about the equation (2.5.5). I'll describe the problem briefly here, but in any case, here's that page from Weinberg's book (page 64)...

Hello. I have a question that has been on my mind for some time. I always see in mathematical physics books that they identify elements of the Lie algebra with group elements "sufficiently close" to the identity. I have never seen a real good proof of this so went on an gave a proof. Let Xi be...

In M.A. Armstrongs book "Groups and Symmetry" in Chapter 12 he introduces the reader to the fascinating topic of Braid Groups. Does anyone know of a book at undergraduate level (or even a popular book) that deals with Braid Groups Can you progress with Braid Groups if you lack a...

I am reading Dummit and Foote on Representation Theory CH 18 I am struggling with the following text on page 843 - see attachment and need some help. The text I am referring to reads as follows - see attachment page 843 for details \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v +...

Homework Statement Let m and n be relatively prime positive integers. Show that if there are, up to isomorphism, r abelian groups of order m and s of order n, then there are rs abelian groups of order mn. Homework Equations The Attempt at a Solution I'm not sure how to go about...

Homework Statement Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n) Homework Equations if two groups form an isomorphism they must be onto, 1-1, and preserve the operation. The Attempt at a...

Hello everyone, I am new to this forum. Need help with this problem How many ways you can select 3-person groups from a group of 8 students? My solution: ---------------- Number of ways to make one group of 3 persons = 8C3 How do I proceed from here? Thank you.

Symmetry, Groups, Algebras, Commutators, Conserved Quantities OK, maybe this is asking too much, hopefully not. I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given. If I understand what I'm reading, there...

Homework Statement Let p be prime and let b_1 ,...,b_k be non-negative integers. Show that if : G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k} then the integers b_i are uniquely determined by G . (Hint: consider the kernel of the homomorphism f_i :G \to...

Linear Algebra Preliminaries in "Finite Reflection Groups In the Preliminaries to Grove and Benson "Finite Reflection Groups' On page 1 (see attachment) we find the following: "If \{ x_1 , x_2, ... x_n \} is a basis for V, let V_i be the subspace spanned by \{ x_1, ... , x_{i-1} ...

I am reading Grove and Benson's book on Finite Reflection Groups and am struggling with some of the basic linear algebra. Some terminology from Grove and Benson: V is a real Euclidean vector space A transformation of V is understood to be a linear transformation The group...

which is the best leaving group? Cl- enolate NR3 I couldn't figure out how to draw the enolate here, but it is a basic 3 carbon one. I believe the answer is Cl because resonance on the enolate will disperse the neg. charge. Am I on the right track or is there a factor I am missing?

Homework Statement Which of following statements are TRUE or FALSE. Why? In any group G with identity element e a) for any x in G, if x2 = e then x = e. b) for any x in G, if x2 = x then x = e. c) for any x in G there exists y in G such that x = y2. d) for any x, y in G there exists z...

I have a question where it says prove that G \cong C_3 \times C_5 when G has order 15. And I assumed that as 3 and 5 are co-prime then C_{15} \cong C_3 \times C_5 , which would mean that G \cong C_{15} ? So every group of order 15 is isomorohic to a cyclic group of order 15...

I'm looking for an algorithm for dividing a set of numbers into groups, and then doing it again in such a way that no numbers are in a group together more than once. For instance if you have 18 numbers and divide them into groups of three you should be able to do this 8 times without any...

Homework Statement The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so...

The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so I am pretty lost in...