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Math Help - How to easily retrieve operation table of group elements from this Cayley diagram?

  1. #1
    Senior Member x3bnm's Avatar
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    How can I retrieve operation table of group elements from this Cayley diagram?

    There's a math problem in Pinter's Abstract Algebra book on pp. 52(exercise G):


    Exercise:

    Every finite group may be represented by a diagram known as a Cayley diagram. A Cayley diagram consists of points
    joined by arrows.

    There is one point for every element of the group.
    The arrows represent the result of multiplying by a generator.

    For example, if G has only one generator a (that is, G is the cyclic group \langle a \rangle), then the arrow \rightarrow
    represents the operation "multiply by a":

    e\rightarrow a \rightarrow a^2 \rightarrow a^3 \rightarrow \cdots

    If the group has two generators, say a and b, we need two kinds of arrows, say dotted arrow and
    line arrow with no dots where dotted arrow means "multiply by a" and lined arrow means "multiply by b".


    For example, the group G = \{ e, a, b, b^2, ab, ab^2 \} where a^2 = e, b^3 = e, and  ba = ab^2 has the following
    Cayley diagram(Figure 1):

    How to easily retrieve operation table of group elements from this Cayley diagram?-cayley_diag2.png


    Moving in the forward direction of the lined arrow means multiplying by b,

    x \rightarrow xb

    whereas moving in the backward direction of the lined arrow means multiplying by b^{-1}:

    x \leftarrow xb^{-1}

    (Note that "multiplying x by b" understood to mean multiplying on the right by b:
    it means xb, not bx) It is also a convention that if a^2 = e(hence a = a^{-1}), then no
    arrowhead is used:

    x\;\;............\;\;xa

    for if a = a^{-1}, then multiplying by a is the same as multiplying by a^{-1}

    The Cayley diagram of a group contains the same information as the group's table. For instance, to find the product (ab)(ab^2)
    in the previous figure 1, we start at ab and follow the path corresponding to ab^2 (multiplying by a, then
    by b, then again by b), which is (Figure 2)

    How to easily retrieve operation table of group elements from this Cayley diagram?-cayley_diag3.png


    This path leads to b, hence (ab)(ab^2) = b

    As another example, the inverse of ab^2 is the path which leads from ab^2 back to e.
    We note instantly that this is ba.

    A point-and-arrow diagram is the Cayley diagram of a group iff it has the following two properties:

    1) For each point x and generator a, there is exactly one a-arrow ending at x;
    furthermore, at most one arrow goes from x to another point y.

    2) If two different paths starting at x lead to the same destination, then those two paths, starting at any point y, lead
    to the same destination.

    Cayley diagrams are a useful way of finding new groups.


    Problem:

    Write the table of the groups having the following Cayley diagram (Remark: You may take any point to represent e(neutral element), because
    there is perfect symmetry in a Cayley diagram. Choose e, then label the diagram and proceed.)


    How to easily retrieve operation table of group elements from this Cayley diagram?-cayley_diag.png


    My problem is: Is it possible to kindly help me find this cyclic group's operation table? How do I choose the elements from this diagram?
    Last edited by x3bnm; November 17th 2012 at 04:23 PM.
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    Re: How to easily retrieve operation table of group elements from this Cayley diagram

    Hey x3bnm.

    Consider that the Cayley graph tells how you multiply one element by another returns a mapped result indicated by the node, the edge, and the arrow associated with those.

    Just take that information and put it into a table and that is your group operation table noting the order of the multiplication.
    Thanks from x3bnm
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  3. #3
    Senior Member x3bnm's Avatar
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    Re: How to easily retrieve operation table of group elements from this Cayley diagram

    Quote Originally Posted by chiro View Post
    Hey x3bnm.

    Consider that the Cayley graph tells how you multiply one element by another returns a mapped result indicated by the node, the edge, and the arrow associated with those.

    Just take that information and put it into a table and that is your group operation table noting the order of the multiplication.
    Thanks chiro for help. I solved the problem. Those who're interested read on:

    How to easily retrieve operation table of group elements from this Cayley diagram?-cayley_diag4.png

    where:
    b^3 = e

    ab^2a = bab

    b^2aba = bab^2

    aba = b^2ab^2

    a^2 = e
    Last edited by x3bnm; November 17th 2012 at 06:03 PM.
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