Exercises

1. Prove that for $n > 2$, $S_n$ has a trivial center, i.e., $Z(S_n) = \{1\}$.

   (HINT: Suppose $f \in Z(S_n)$ and $f \not= 1$. Write $f$ in its disjoint cycle form (see equation (3.2) and Theorem 3.1.2). Consider the following three cases:

   Case 1

   $f$ has at least one $m$-cycle with $m \geq 3$. Without loss of generality, assume $(a_1,a_2,...,a_k)$ in (3.2) is such that $k \geq 3$. Then calculate $f\cdot (a_1,a_2)$ and $(a_1,a_2)\cdot f$.

   Case 2

   the disjoint cycle decomposition of $f$ has at least two transpositions ($2$-cycles), say $k = j = 2$ in (3.2), i.e., $f = (a_1,a_2)(b_1,b_2)...$. Then calculate $(a_1,b_1,a_2)\cdot f$ and $f\cdot (a_1,b_1,a_2)$.

   Case 3

   $f = (a_1,a_2)$; then calculate $(a_1,a_2,a_3)f$ and $f(a_1,a_2,a_3)$ - recall $n \geq 3$.

   In each case $a_1, a_2, a_3, b_1, b_2 \in \{1, 2, ..., n\}$ and we can ``calculate'' all we need to know by computing the effect of each permutation (remember permutation is a function) acting on $a_1$.)

2. (a) Suppose $f,g \in S_n$ and they have the same cycle structure. Prove $f \sim g$ ($\sim$ means the relation of being conjugate).

   (b) Explain why there are as many conjugacy classes in $S_n$ as there are partitions of $n$. (HINT: Do it for $n = 3$, $S_3$, first!)

3. In $S_5$ perform the indicated operations. Write the result first as a product of disjoint cycles and then in the 2-row form (2.1):

   (a) $(1,2,3) (1,3,5) (2,4) (1,2,3)^{-1}$,

   (b) $(1, 3,4,5,2) (1,2) (3,5) (1,3,4,5,2)^{-1}$.

4. In $S_4$, determine the number of conjugacy classes and the number of permutations in each class. (See problem 2 for Section 3.1.)