group

  1. A

    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. B

    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. topsquark

    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. D

    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. E

    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. E

    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. C

    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. E

    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. E

    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. H

    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. O

    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. E

    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. M

    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. E

    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. S

    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. Y

    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. Y

    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. M

    Allowances of the group

    Hello After this, How do I calculate the allowances of the group?
  19. E

    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. H

    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.