Actions and Sylow theorems

Concepts

• Symmetric group 10 exercisesThe symmetric group S_n is the set of bijections σ of \1,dots,n\ onto itself, with composition στ as the operation.
• Cycle of a permutation 10 exercisesA cycle of length k in S_n is a permutation sending a_1mapsto a_2mapstocdotsmapsto a_kmapsto a_1 and fixing all other elements.
• Sign of a permutation 10 exercisesThe sign sgn(σ) of a permutation σ is +1 or -1 according to whether σ is a product of an even or odd number k of transpositions.
• Group action 10 exercisesAn action of a group G on a set X assigns to each g,h∈ G and x∈ X an element g· x∈ X, so that the identity e moves nothing and th…
• Orbit-stabilizer theorem 10 exercisesFor a finite group G acting on X and an element x∈ X, the size of the orbit of x is the index of its stabilizer in G.
• Orbit and stabilizer 10 exercisesIf a group G acts on a set X, the orbit of x∈ X is the set of all its images g· x, and its stabilizer is the subgroup of elements…
• p-group 10 exercisesA p-group is a finite group G whose order is a power p^n, with n≥1, of a prime number p.
• Sylow theorems 10 exercisesA finite group G of order p^k m, with p prime and pnmid m, has subgroups of order p^k, all conjugate to each other; their number…