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. ...continues

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 the ...continues

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 ...continues

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 orde ...continues

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.

Fermat and Wilson's Theorem

1. If and are distinct primes prove that for any integer a, Use Fermat’s theorem 2. Show that if and are both primes, then 4[ (mod Use Wilson’s theorem. 3. Let be an odd prime. Prove that if g is primitive root modulo and (mod is not Use the binomial expansion See attached file for full ...continues

Find the simultaneous solutions of the following congruences

1. Prove that gcd (a, lcm[b, c]) = lcm[gcd(a,b), gcd(a,c)]. 2. Find the simultaneous solutions of the following congruences: 2x ≡ 1(mod 5) 3x ≡ 9 (mod 6) 4x ≡ 1 (mod 7) 5x ≡ 9 (mod 11)

Properties of groups

If G is a group, then (1) the identity element of G is unique, (2) every a belongs to G has a unique inverse in G.

Show that an r-cycle is an even permutation if and only if r is odd.

1. If alpha is an r-cycle, show that alpha^r = (1). [There's a hint that If alpha = (i sub 0 ... i sub r-1), show that alpha ^k(i sub 0) = i sub k.] 2. Show that an r-cycle is an even permutation if and only if r is odd. 3. If alpha is an r-cycle and 1

Noncyclic Group Order 4.

Noncyclic Group Order 4. See attached file for full problem description.