2. Prove that stabilizers of different elements of the same orbit are conjugated subgroups in G.
3. In a group G, let Zg denotes the subgroup consisting of all those elements which commute with g (it is called the centralizer of g). Prove that the number of classes of conjugate elements in G is equal to the average size of the centralizer: sumg |Zg| /|G|.
Hint: Apply Cauchy's counting principle - called Burnside's formula in Fraleigh's book - to the action of G on itself by conjugations.
4. Compute the number of different necklaces formed by 17 black or white beads of the same spherical shape and size threaded on a circular thread. (The necklaces which can be obtained from each other by rotating and/or flipping them in space are considered the same.)
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.