# 52 only Section 9Orbits, Cycles, and the Alternating Groups87(a * b) * c of the associative property for G under * were written on the wall with a magic marker. Whatwould a person see when looking at the other side of the wall from the next room in front of yours?b. Show from the mathematical definition of *' that G' is a group under *'.52. Let G be a group. Prove that the permutations o,:G → G, where pa(x) = xa for a e G and x e G, do forma group isomorphic to G.53. A permutation matrix is one that can be obtained from an identity matrix by reordering its rows. If P is ann x n permutation matrix and A is anyn x n matrix and C = PA, then C can be obtained from A by makingprecisely the same reordering of the rows of A as the reordering of the rows which produced P from I.a. Show that every finite group of order n is isomorphic to a group consisting of n x n permutation matricesunder matrix multiplication.b. For each of the four elements e, a, b, and c in the Table 5.11 for the group V, give a specific 4 x 4 matrixthat corresponds to it under such an isomorphism.SECTION 9 ORBITS, CYCLES, AND THE ALTERNATING GROUPSOrbitsEach permutation o of a set A determines a natural partition of A into cells withbe A are in the same cell if and only if b = o"(a) for some n e Zpropertythatа,establish this partition using an appropriate equivalence relation:For a, b e A, let a ~ b if and only if b = o"(a) for some n e Z.hy Condition (1) is indeed an equivalence relation.

Name: # 52 only Section 9Orbits, Cycles, and the Alternating Groups87(a * b)
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Description: # 52 only Section 9Orbits, Cycles, and the Alternating Groups87(a * b) * c of the associative property for G under * were written on the wall with a magic marker. Whatwould a person see when looking at the other side of the wall from the next room in