Can someone please help with these four questions? I am totally lost on what i a
ID: 1719421 • Letter: C
Question
Can someone please help with these four questions? I am totally lost on what i am suppose to do.
a) Let G act on X, and let S, T X. Prove that GST = GS GT .
b) Let G act on X, and let S X. Recall that GS = {g G | s S gs = s}. Prove that GS G.
c) You are to color each of the four vertices of a square with any of six different colors. Two colorings are equivalent if one can be obtained from the other by any symmetry of the square (not just rotation). How many non-equivalent colorings are there?
d) You are to color each of the inner and outer vertices of the star used in problems 1 through 3 with any of four different colors. Two colorings are equivalent if one can be obtained from the other by any symmetry of the star. How many non-equivalent colorings are there?
Explanation / Answer
a) By definition GSUT = {g in Gs SUT gs = s}
= {g Gs S gs = s , s T gs = s }
= {g Gs S gs = s} and {g Gs T gs = s}
= GS GT
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b) GS = {g G | s S gs = s}
G = Entire G
Hence G will contain all elements of G including GS
Or if gs =s, since S is a subset gs belongs to G also
i.e. GS is contained in G
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c) 4 vertices with 6 different colours
A square has 4 symmetric lines, 2 diagonals and 2 lines joining mid points.
Hence each vertex should have different colour to avoid symmetry
Hence non equivalent colourings are 6P4 = 6x5x4x3 = 360
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d) A start with 6 vertices (inner) and 6 outer has symmetries as
6 symmetrical lines are there 3 joining inner vertices and 3 joining outer vertices