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

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 homomorphism, determine the Kernel: G is the group of non-zero real numbers under multiplication, ¯G = G, φ(x) = x^2 all xєG.

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 is homomorphism, determine the Kernel: G is the group of non-zero real numbers under multiplication, ¯G = G, φ(x) = 2^x all xєG.

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 case in which it is homomorphism, determine the Kernel: G is any abelian group and ¯G = G, φ(x) = x^5 all xєG.

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 element in G. Define φ:G→ G by φ(x) = gxg^-1 Prove that φ is an isomorphism of G onto G.