Group Theory (CV): If N is a normal subgroup of G and aЄN show that every conjugate of a in G is also in N .

Modern Algebra Group Theory (CV) Permutation Groups Another Counting Principle If N is a normal subgroup of G and aЄN show that every conjugate of a in G is also in N .

Group Theory (CVI): Prove that O(N) = Σ ca for some choices of a in N.

Modern Algebra Group Theory (CVI) Permutation Groups Another Counting Principle Prove that O(N) = Σ ca for some choices of a in N.

Group Theory (CVII): Using O(N) = Σca for some choices of a in N , prove that in A5 there is no normal subgroup N other than (e) and A5 .

Modern Algebra Group Theory (CVII) Permutation Groups Another Counting Principle Using O(N) = Σca for some choices of a in N , prove that in A5 there is no normal subgroup N other than (e) and A5 .

Group Theory (CVIII): Permutation Groups: Another Counting Principle: Using the theorem ‘ If O(G) = p^n , where p is a prime number, then Z(G) ≠ (e) ’ as a tool, prove that if O(G) = p^n where p is a prime number, then G has a subgroup of order p^α for all 0 < α ≤ n .

Modern Algebra Group Theory (CVIII) Permutation Groups Another Counting Principle Using the theorem ‘ If O(G) = p^n , where p is a prime number, then Z(G) ≠ (e) ’ as a tool, prove that if O(G) = p^n where p is a prime number, then G has a subgroup of order p^α for all 0 < α ≤ n .

Group Theory (CIX): Sylow’s Theorem: In the symmetric group of degree 4, S4 , find a 2-Sylow subgroup and a 3-Sylow subgroup.

Modern Algebra Group Theory (CIX) Sylow’s Theorem In the symmetric group of degree 4, S4 , find a 2-Sylow subgroup and a 3-Sylow subgroup.