Group Theory (LXX): If G is a group, then Aut(G), the set of automorphisms of G, is also a group.

Modern Algebra Group Theory (LXX) The Set of all Automorphisms of a Group If G is a group, then Aut(G), the set of automorphisms of G, is also a group.

Group Theory (LXXI): Let φ be a homomorphism of G onto G¯ with kernel K . Then G/K ≈ G¯

Modern Algebra Group Theory (LXXI) Homomorphism of a Group The Kernel of a Homomorphism Let φ be a homomorphism of G onto G¯ with kernel K . Then G/K ≈ G¯

Group Theory (LXXII): Let G be a group and Z(G), the centre of G, then G/Z(G) ≈ I(G) , where I(G) is the set of all inner automorphisms of G.

Modern Algebra Group Theory (LXXII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group Let G be a group and Z(G), the centre of G, then G/Z(G) ≈ I(G) , where I(G) is the set of all inner automorphisms of G.

Group Theory (LXXIII): Let G be a group, consider the mapping of G into itself, λg, defined for gЄG by x λg =gx for all xЄG.Prove that λg is one-to-one and onto, and that λgh = λhλg.

Modern Algebra Group Theory (LXXIII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group Let G be a group, consider the mapping of G into itself, λg, defined for gЄG by xλg =gx for all xЄG. Prove that λg is one-to-one and onto, and that λgh = λhλg

Group Theory (LXXIV): Let G be a group, consider the mapping λg :G→G defined for gЄG by xλg = gx for all xЄG and the mapping τg :G→G defined for xЄG by xτg = xg for every xЄG. Prove that for any g,hЄG, the mappings λg , τh satisfy λg τh = τh λg .

Modern Algebra Group Theory (LXXIV) The Set of all Automorphisms of a Groupn The Set of all Inner Automorphisms of a Group Let G be a group, consider the mapping λg :G→G defined for gЄG by xλg = gx for all xЄG and the mapping τg :G→G defined for xЄG by xτg = xg for every xЄG. Prove that for any g,hЄG, the mappings λg , τh satisfy λg τh = τh ... click for more