Group Theory (CX): Sylow’s Theorem: Find all 3-Sylow subgroups of or, Sylow 3-subgroups and 2-Sylow subgroup or, Sylow 2-subgroups of the symmetric group of degree 4, S4.

Modern Algebra Group Theory (CX) Sylow’s Theorem Find all 3-Sylow subgroups of or, Sylow 3-subgroups and 2-Sylow subgroup or, Sylow 2-subgroups of the symmetric group of degree 4, S4.

Group Theory (CXI): Permutation Groups: Another Counting Principle:If O(G) = pn a prime number , prove that there exists subgroups Ni ( for some r) such that G = N0 > N1 >N2 >N3 > … >Nr = (e) where Ni is a normal subgroup of Ni-1 and where Ni-1 /Ni is abelian. Here Ni-1>Ni means Ni-1 is superset of Ni.

Modern Algebra Group Theory (CXI) Permutation Groups Another Counting Principle If O(G) = pn a prime number , prove that there exists subgroups Ni ( for some r) such that G = N0 > N1 >N2 >N3 > … >Nr = (e) where Ni is a normal subgroup of Ni-1 and where Ni-1 /Ni is abelian . Here Ni-1>Ni means Ni-1 is superset of Ni.

Group Theory (CXII): Groups of Order Power of a Prime: If O(G) = p^n , p a prime number , and H ≠ G is a subgroup of G, show that there exists an x Є G, x does not belong to H such that x^(-1)Hx = H.

Modern Algebra Group Theory (CXII) Groups of Order Power of a Prime Another Counting Principle If O(G) = p^n , p a prime number , and H ≠ G is a subgroup of G, show that there exists an x Є G, x does not belong to H such that x^(-1)Hx = H.

Group Theory (CXIII): Prove that a group of order 108 must have a normal subgroup of order 9 or 27.

Modern Algebra Group Theory (CXIII) Groups of Order having Power of a Prime Another Counting Principle Prove that a group of order 108 must have a normal subgroup of order 9 or 27.

In this posting, the question is to prove that every permutation in an alternate group is the product of n-cycles.

Prove that every permutation in an alternate group is the product of n-cycles.