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Math Help - Alternating Group A4 and Cayley diagraph

  1. #1
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    Alternating Group A4 and Cayley diagraph

    Show that A4 = {(12)(34), (123)}, and draw a picture of the Cayley graph with
    respect to this generating set.

    I know that the elements generated by (12)(34) and by (123) are all in A4, but how can I show that there are 12 distinct elements, therefore showing that A4 is generated by those two cycles above?

    Thank you.
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  2. #2
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    Re: Alternating Group A4 and Cayley diagraph

    Hi,
    Set H=<(12)(34),(123)>. Then H=A4: 2 and 3 both divide the order of H but since A4 has no subgroup of order 6 (if it did a Sylow 3 subgroup of A4 would be normal in A4), H=A4.

    Here's a Cayley diagram for A4 and the given generators. The products of cycles are formed "left to right" and the elements are post multiplied by the generators. If you prefer products "right to left", just think of the elements being pre multiplied by the generators.

    Alternating Group A4 and Cayley diagraph-mhfcayleydiagrama4.png
    Thanks from christianwos
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  3. #3
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    Re: Alternating Group A4 and Cayley diagraph

    Thank you so much!
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