Presentation of a group to generators in A(S)

In summary, presenting a group to generators in A(S) allows for a more structured and simplified representation of the group, aiding in understanding its properties and operations. This is done by defining a set of generators and a set of relations that determine the group's structure. It differs from presenting a group as a Cayley graph, which is a visual representation. Presenting a group to generators can help in solving problems by applying algebraic techniques, but there are limitations when dealing with large numbers of generators and complex relations, and not all groups can be presented in this way.
  • #1
TylerH
729
0
Is there a general algorithm for taking the presentation of a group and get the permutation generators for the subgroup of A(S) to which the group is isomorphic?

For example, given x^5=y^4=e, xy=f(c^2) how do I find (12345) and (1243), the permutations corresponding to x and y? BTW, the example is the Frobenious group of order 20, but I'm asking about a general method.
 
Physics news on Phys.org
  • #2
I think the usual method is the Todd-Coxeter algorithm. It's covered quite extensively in Artin's Algebra.
 

1. What is the significance of presenting a group to generators in A(S)?

Presenting a group to generators in A(S) is a way to represent a group in a more structured and simplified form. It allows us to understand the group's properties and operations better.

2. How is a group presented to generators in A(S)?

A group is presented to generators in A(S) by defining a set of generators and a set of relations or rules that these generators must follow. These relations determine the group's structure and its elements' relationships with each other.

3. What is the difference between presenting a group to generators and presenting it as a Cayley graph?

Presenting a group to generators is a more abstract and algebraic way of representing a group, while presenting it as a Cayley graph is a visual representation. In a Cayley graph, the generators are represented as vertices, and the relations between them are shown as edges.

4. How can presenting a group to generators help in solving problems related to the group?

Presenting a group to generators can help in solving problems related to the group by providing a clear understanding of the group's structure and its elements' relationships. It allows us to apply algebraic techniques and properties to solve problems efficiently.

5. Are there any limitations to presenting a group to generators in A(S)?

Yes, there are limitations to presenting a group to generators in A(S). It can become challenging to determine the structure of a group when presented with a large number of generators and complex relations. Also, not all groups can be presented to generators in A(S).

Similar threads

I Isomorphisms between C4 & Z4 Groups
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Show that the groups are abelian
    • Linear and Abstract Algebra
Replies
1
Views
784
I Finding All Automorphisms of Group
    • Linear and Abstract Algebra
Replies
2
Views
1K
I The quotient group of a group with a presentation
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB *elements and generators of U(14)
    • Linear and Abstract Algebra
Replies
2
Views
5K
MHB The group is isomorphic to one of the groups
    • Linear and Abstract Algebra
Replies
9
Views
2K
I Meaning of terms in a direct sum decomposition of an algebra
    • Linear and Abstract Algebra
Replies
8
Views
2K
Showing any transposition and p-cycle generate S_p
    • Linear and Abstract Algebra
Replies
1
Views
4K
Determining if Subset is a Subgroup by using Group Presentation
    • Linear and Abstract Algebra
Replies
2
Views
2K
Discussing the mathematical formalism of generators (Lorentz Group)
    • Quantum Physics
Replies
31
Views
3K

Hot Threads

Recent Insights

Back
Top