# group

1. ### Presentation of a group in standard form

Hi I have the following group presentation G = a,b,c | ab = ba, ac = ca, bc = cb, a^2 = b^3 = c^3 = e. I need to show that all the elements of G can be written in the standard form a^ib^jc^k for i ∈{0,1} and j,k ∈{0,1,2}. I'm not really sure how to go about this so any help would be great...
2. ### Conjugation in the Free Group

Let F_n denote the (nonabelian) free group on the n generators u_1,...,u_n, and let h \in F_n be arbitrary. My question is, does there exist a g \in F_n such that g^{-1} h g = h, besides g=e (the identity); is such an equation in the free group possible? Obviously this equation would imply they...
3. ### Group Theory Project

My research into String Theory has stalled. I'm stuck at Conformal Field Theory. The problem is that I have taught myself all the advanced Math that I know and I know there are gaps, omissions, and outright errors in my knowledge. What I need is a good tutor for several subjects and I'm not...
4. ### Group or not ?

Here is the question, the definition is confusing me: https://gyazo.com/e06f9ed8257127747ba60d2ebd77fed8 My attempt: (i) Holds since a+(b+c)=(a+b)+c and ax(bxc)=(axb)xc (ii)holds since if e=0 a+0=0=0+a=a. if e=1 a1=1a=a (iii) a+b =b+a but not equal to zero since zero has no invest; axb=bxa...
5. ### Group theory: direct products.

Hi: Let G be a p-group, p prime, G= A1 x ..., x Ak, Ai cyclic, H <= G, f: G --> G/H the natural homomorphism, Ai= <x_i>, |<x_i>| = p^{a_i}. If G/H= B1 x ... x Bm, with Bi= <y_i>, |<y_i>| = p^{b_i}, and a_m <= b_k. I say that for all i there exists j such that f(Ai) <= Bj. Is this true, because...
6. ### Group theory: minimal generating sets.

I want to know: In an finiteAbelian group, is a minimal generating set necessarily an independent set? I have tried to prove it but I have failed. Perhaps if the group is primary (a p-group for some prime p)? More precisely, suppose I have a set that generates G. By removing elements from the...
7. ### Group Theory

Hi, I'm studying Group Theory, and I've come across this problem. Say I have a subgroup K which is not normal, how do you determine other two subgroups of the general group say L to which the subgroup K is conjugate. Thanks for the help in advance. And i really appreciate this forum. caboampong
8. ### What are the elementary divisors of U(Z/nZ) (group theory)?

Hi: The problem: What are the elementary divisors of U(n), the multiplicative group of all congruence classes [a] mod n with (a,n)=1? Notation: a = b means a equal b or a isomorphic to b depending on context. I find that U(p^m q^n) = U(p^m) x U(q^n), where p,q primes, using the Chinese...
9. ### Extending an independent set in an abelian finite group.

Hi: Trying to prove that an abelian finite group G with a subgroup H has a subgroup isomorphic with G/H, I stumbled on the following problem: let G be finite abelian and supose A= {a_1, a_2} is an independent subset of G which does not generate G. Can I find a_3 such that {a_1, a_2, a_3} is...
10. ### Number of group homomorphisms

Hi, Count the number of homomorphisms from the Euler group U8 to the symmetric group S5.Show that there are no monomorphisms. Thank's in advance.
11. ### Group Definition

Hello, I'm new to the forum and I'm taking a linear algebra class with a very hard professor. My question is this. From the course textbook (Jean Gallier 2013, Basics of Algebra, Topology, and Differential Calculus) it says; "The set R of real numbers has two operations + : RXR-->R...
12. ### Mathematical anomaly (group theory).

Let f: G --> G/K, where K normal in G, f the natural homomorphism. Let H in G, K not in H. What is f(H)? Well, f(H)= {f(h): h in H}= {Kh: h in H}= H/K. What is this monster H/K?
13. ### Group R^× isomorphic to the group R?

Question: Is the group R^× isomorphic to the group R? Why? R^× = {x ∈ R|x not equal to 0} is a group with usual multiplication as group composition. R is a group with addition as group composition.Is there any subgroup of R^× isomorphic to R? What I Know: Sorry, I would have liked to show some...
14. ### If p and q are primes then every group of order p^m q^n is solvable.

Hi: Burnside proved (using representation theory) that the number of elements in a conjugacy class of a finite simple group can never be a prime power larger than 1. Use this fact to prove Burnside's theorem: If p and q are primes then every group of order p^m q^n is solvable. What I did: let...
15. ### first homolgy group of a disk with n holes

Hi; Let D^2 be a 2-dim disk with n holes, i.e D^2\(S^0 * D^2\$)^n. Then is it true that the first homology group of this space is Z^n. Thank you in advance
16. ### sylow p group

Suppose G is finite group and P is a sylow p group of G. N is normal group of G. is the intersection of P and N is Sylow-p group of N?
17. ### Abelian group

Dear all: Can you prove the following theorem for me? I am not sure if it is true or not Suppose G is a finite group and H is a normal group in G. And we define the natural homomorphism f:g------>gH if in quotient group G/H, there exist a abelian subgroup N/H. Can we prove f^(-1)(N/H) is...
18. ### Allowances of the group

Hello After this, How do I calculate the allowances of the group?
19. ### Elementary group theory problem.

Let G be a group of order k 2^m, where k is odd. Prove that if G contains an element of order 2^m, then the set of all elements of odd order in G is a (normal) subgroup of G. (Hint. Consider G as permutations via Cayley's theorem, and show that it contains an odd permutation.) I've attempted...
20. ### Showing that there is no simple group of order 6p^r

Show that there is no simple group of order 6p^r for any prime p and positive integer m.