Group Theory (XLVIII): If H is a subgroup of G, let N(H) = {gєG|gHg^-1 = H}. Prove that H is normal in G if and only if N(H) = G.

Modern Algebra Group Theory (XLVIII) Normal Subgroups of a Group Normalizer of a Subgroup of a Group Centralizer of a Subgroup of a Group If H is a su ...continues

Group Theory (L):Homomorphism of a Group: Verify if the mapping defined is a homomorphism and in that case in which it is homomorphism,determine the Kernel:G is the group of non-zero real numbers under multiplication, ¯G = G, φ(x) = x^2 all xєG.

Modern Algebra Group Theory (L) Homomorphism of a Group Kernel of the Homomorphism Verify if the mapping defined is a homomorphism and in that case in which it is hom ...continues

Group Theory (LI): Homomorphism of a Group: Verify if the mapping defined is a homomorphism and in that case in which it is homomorphism,determine the Kernel:G is the group of non-zero real numbers under multiplication,¯G = G, φ(x) = 2^x all xєG.

Modern Algebra Group Theory (LI) Homomorphism of a Group Kernel of the Homomorphism Verify if the mapping defined is a homomorphism and in that case in which it ...continues

Group Theory (LIV): Homomorphism of a Group: Verify if the mappings defined is a homomorphism and in that case in which it is homomorphism,determine the Kernel:G is any abelian group and ¯G = G, φ(x) = x^5 all xєG.

Modern Algebra Group Theory (LIV) Homomorphism of a Group Kernel of the Homomorphism Verify if the mappings defined is a homomorphism and in that ...continues

Group Theory (LV): Let G be any group, g a fixed element in G. Define φ:G→ G by φ(x) = gxg^-1.Prove that φ is an isomorphism of G onto G.

Modern Algebra Group Theory (LV) Isomorphism of a Group Automorphism of a Group Inner Automorphism of a Group Let G be any group, g a fixed ...continues

Group Theory (LVI): Prove that the centre of a group is always a normal subgroup.

Modern Algebra Group Theory (LVI) Normal Subgroups of a Group Centre of a Group Prove that the centre of a group is a ...continues

Group Theory (LVII): A group of order p^2,where p is a prime number, is abelian.

Modern Algebra Group Theory (LVII) A Group of Order p^2, where p is a prime Normalizer or Centralizer of an element of a Group Abelian Group ...continues

Group Theory (LVIII): Prove that a group of order 9 is abelian.

Modern Algebra Group Theory (LVIII) A Group of Order p^2, where p is a prime Normalizer or Centralizer of an element of a Group Abelian Group ...continues

Group Theory (LIX): Quotient Group or Factor Group: If G is abelian and if N is any subgroup of G, prove that G/N is abelian.

Modern Algebra Group Theory (LIX) Quotient Group or Factor Group Abelian Group If G is abelian and if N is any subgroup of G, prove that ...continues

Group Theory (LXI): Automorphism of a Group: Is the mapping given below an automorphism of the group ? G group of integers under addition, T:x→ -x

Modern Algebra Group Theory (LXI) Automorphism of a Group Is the mapping given below an automorphism of the group ? G group of integers under addition, T:x→ -x