Abstract Algebra Problems

1. Given a left action of a group G on a set M, let x be an element of M, and H={ g | gx=x } the stabilizer of x. Prove that the map G->M: g |-> gx establishes a bijection between the set G/H of left cosets and the orbit of x.
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.)

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