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There are two orbits with more than one element. • The permutation τ = 1 2 3 4 5 6 7 1 3 5 4 7 6 2 is a cycle. The permutation has the orbit {2, 3, 5, 7} and all other elements are fixed. • The identity permutation ι is a cycle. Every element is fixed by this permutation, so every orbit has one element. ☐ The notation for permutations that we have been using is called the tableau notation. Although this notation has its merits, it complicates some matter. We will introduce a different notation for a permutation.

We say that x y if there exists an integer k such that x = y + 5k . We see that x x since x = x + 5 · 0 . If x y then x = y + 5k for some integer k . By basic algebra we have y = x + 5(−k) . Since −k is an integer this shows that y x . If x y and y z then there exist integers k, m such that x = y + 5k and z = y + 5m . We again use some basic algebra to see that x = (z − 5m) + 5k = z + 5(k − m) . Since k − m ∈ Z we see that x z. ☐ This last example explains why we can consider −4, 6 and 101 to be the same.

While it is true that 1 · 2 = 1 , the number 12 is not an integer, and cannot be used as an inverse element of 2. 2 4. n × n matrices with real entries and nonzero determinant under matrix multiplication is a nonabelian group. Matrix multiplication is associative. The identity matrix In , a matrix with entry of 1 along the diagonal and zeros elsewhere, has the property that A · In = In · A for all n × n matrices A . Since any matrix A in this set has nonzero determinant, there is an inverse matrix A−1 .

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