Question

1. Let $alpha = egin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \ 2 & 1 & 6 & 4 & 3 & 5 end{bmatrix}$ and $eta = egin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \ 3 & 1 & 6 & 2 & 4 & 5 end{bmatrix}$. Write the following permutations as a product of disjoint cycles: (i)$alpha^{-1}$ (ii)$alphaeta$ (iii) $eta^3$ 2. Let $alpha = (125)(1246)$. Write $alpha^{38}$ as a product of disjoint cycles. 3. Let $alpha$ and $eta$ be elements of $S_n$. Prove that $alpha^{-1}eta^{-1}alphaeta$ is an even permutation. 4. Show that for $n ge 3$, $Z(S_n) = {varepsilon}$ 5. Find all integers $n$ such that $S_6$ contains an element of order $n$.

          1. Let $alpha = egin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \ 2 & 1 & 6 & 4 & 3 & 5 end{bmatrix}$ and $eta = egin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \ 3 & 1 & 6 & 2 & 4 & 5 end{bmatrix}$. Write the following permutations as a product of disjoint cycles: (i)$alpha^{-1}$ (ii)$alphaeta$ (iii) $eta^3$

2. Let $alpha = (125)(1246)$. Write $alpha^{38}$ as a product of disjoint cycles.

3. Let $alpha$ and $eta$ be elements of $S_n$. Prove that $alpha^{-1}eta^{-1}alphaeta$ is an even permutation.

4. Show that for $n ge 3$, $Z(S_n) = {varepsilon}$

5. Find all integers $n$ such that $S_6$ contains an element of order $n$.

1. Let alpha = eginbmatrix 1 2 3 4 5 6 2 1 6 4 3 5 endbmatrix and eta = eginbmatrix 1 2 3 4 5 6 3 1 6 2 4 5 endbmatrix. Write the following permutations as a product of disjoint cycles: (i)alpha^-1 (ii)alphaeta (iii) eta^3

2. Let alpha = (125)(1246). Write alpha^38 as a product of disjoint cycles.

3. Let alpha and eta be elements of Sn. Prove that alpha^-1eta^-1alphaeta is an even permutation.

4. Show that for n ge 3, Z(Sn) = varepsilon

5. Find all integers n such that S6 contains an element of order n.

Added by Tracy W.

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