In the puzzle shown, six round tiles (labeled 0, 1, 2, 3, 4, infinite) are moved

Question

In the puzzle shown, six round tiles (labeled 0, 1, 2, 3, 4, infinite) are moved around tracks. The group G, consisting of all legal moves, is generated by the two 4-cycles alpha = (0, 1, 4, 3) and Beta = (1, 4, 2, infinite). Note that G is a subgroup of Symf0, 1, 2, 3, 4, infinite), the group of all permutations of the six tiles, a group isomorphic to S_6. a. The 5-cycle around the outer track is gamma = (0, 1, infinite, 2, 3). Express gamma as a combination of the generators a and Beta i.e. a product of powers of the (This shows that gamma sum G = (alpha, beta).) b. Determine the order of G c. How many elements of each order does G have? d. How many elements of G are even permutations, and how many arc odd permutations? e. The group G is isomorphic to a group studied previously this semester. Based on your answers to (b) and (c) above, conjecture which group this is. F. Name another group studied previously this semester, having the same order as G, but which is not isomorphic to G. The information in (c) can be used to justify why these two groups are in fact not isomorphic. You many use Maple or other comparable software in answering this problem. Before starting, you will probably want to rename the tiles 1, 2, 3, 4, 5, 6 to be acceptable for input to Maple. To answer (c) and (d), it is helpful first to list all elements of G, use the Maple command Elements (G);

Explanation / Answer

a)Alpha might be taken as (0,1,a,2,3) itself and beta may be taken as (0,1,a,2,4).The combination(multiplication of both)gives the whole set.   

b)The order of the graph is the total number of vertices in it.Here in this case it is 6.

c)This can be seen to count the number of edges falling on each vertex.

Vertex 1: 01,41,a1........3 edges

Vertex 2: 2a........1 edges

Vertex 3: 03,43........2 edges

Vertex 4: 41,43........2 edges

Vertex 0: 10,30........2 edges

Vertex a: 1a,2a........2 edges

Note:a means infinity symbol

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