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6.E: Pólya–Redfield Counting (Exercises)

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    7235
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    6.2: Groups of Symmetries

    Exercise \(\PageIndex{2.1}\)

    Find the 12 permutations of the vertices of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron. Use the labeling below.

    clipboard_eef51366e515526f833e365c199534998.png
    Figure \(\PageIndex{1}\)
    Exercise \(\PageIndex{2.2}\)

    Find the 12 permutations of the edges of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron. Use the labeling below.

    clipboard_e6f5d057fa6622d8c211cec928e77e6a2.png
    Figure \(\PageIndex{2}\)

    6.3: Burnside's Theorem

    Exercise \(\PageIndex{3.1}\)

    Write the 12 permutations of the vertices of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron in cycle form. Use the labeling below.

    clipboard_e9ed19d98d131b88f20ae3a2f2578fad2.png
    Figure \(\PageIndex{3}\)
    Exercise \(\PageIndex{3.2}\)

    Find the number of different colorings of the vertices of a regular tetrahedron with \(k\) colors, modulo the rigid motions.

    Exercise \(\PageIndex{3.3}\)

    Write the 12 permutations of the edges of the regular tetrahedron corresponding to the 12 rigid motions of the regular tetrahedron in cycle form. Use the labeling below.

    clipboard_e7118dc617752564d10c2910b95676457.png
    Figure \(\PageIndex{4}\)
    Exercise \(\PageIndex{3.4}\)

    Find the number of different colorings of the edges of a regular tetrahedron with \(k\) colors, modulo the rigid motions.

    Exercise \(\PageIndex{3.5}\)

    Find the number of non-isomorphic graphs on 5 vertices "by hand'', that is, using the method of Example 6.3.2.

    6.4: Pólya-Redfield Counting

    Exercise \(\PageIndex{4.1}\)

    Find the cycle index \(P_G\) for the group of permutations of the vertices of a regular tetrahedron induced by the rigid motions. (See Exercise 6.E.3.1 in Section 6.E.)

    Exercise \(\PageIndex{4.2}\)

    Using the previous exercise, write out a full inventory of colorings of the vertices of a regular tetrahedron induced by the rigid motions, with three colors, as in Example 6.4.3. You may use Sage or some other computer algebra system.

    Exercise \(\PageIndex{4.3}\)

    Find the cycle index \(P_G\) for the group of permutations of the edges of \(K_5\). (See Exercise 6.E.3.5 in Section 6.E. Don't use the general formula above.)

    Exercise \(\PageIndex{4.4}\)

    Using the previous exercise, write out a full inventory of the graphs on five vertices, as in Example 6.4.4. You may use Sage or some other computer algebra system.


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