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

Modern Algebra Group Theory (CXII) Groups of Order Power of a Prime Another Counting Principle If ...continues

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

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.

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

Systems of Congruences : Simultaneous Solutions

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)

Groups : Uniqueness of Indentities and Inverses

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.

Permutations and R-Cycles

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

Isomorphic : Noncyclic Group Order 4

Let V be a noncyclic group of order 4. (We know that all such groups are isomorphic, one is given in example 2.96). How large is Aut(V)? To which familiar group is Aut(V) isomorphic?

Mappings, Injective and Surjective Functions and Cycles

1. Let f : X -> Y and g : Y -> Z be mappings. (1) Show that if f and g are both injective, then so is g o f : X -> Z (2) Show that if f and g are both surjective, then so is g o f : X -> Z. 2. Let alpha = 1 2 3 4 5 and Beta = 1 2 3 4 5 3 5 1 2 4 3 2 4 5 1 . ...continues