Group Theory (XXXVII): If N is a normal subgroup of G and H is any subgroup of G, prove that NH is a subgroup of G.

Modern Algebra Group Theory (XXXVII) Subgroups of a Group Normal Subgroups of a Group If N is a normal subgroup of G and H is any subgroup of G, pro ...continues

Group Theory (XXXVIII): To show that the intersection of two normal subgroups of G is a normal subgroup of G.

Modern Algebra Group Theory (XXXVIII) Subgroups of a Group Normal Subgroups of a Group To show that the intersection of two normal subgroups of G is a normal subgroup of G.

Group Theory (XXXIX): If H is a subgroup of G and N is a normal subgroup of G, show that H∩N is a normal subgroup of H.

Modern Algebra Group Theory (XXXIX) Subgroups of a Group Normal Subgroups of a Group If H is a subgroup of G and N is a normal subgroup of G, show that ...continues

Group Theory (XL): Every subgroup of an abelian group is normal.

Modern Algebra Group Theory (XL) Subgroups of an Abelian Group Normal Subgroups of a Group Every subgroup of an abelian group is normal.

Group Theory (XLI): Suppose that N and M are two normal subgroups of G and that N∩M = (e).Show that for any nєN, mєM, nm = mn.

Modern Algebra Group Theory (XLI) Subgroups of a Group Normal Subgroups of a Group Suppose that N and M are two normal subgroups of G and that N∩M = ( ...continues

Group Theory (XLIII): Subgroups of the type gHg^-1: Let G be a group, H a subgroup of G. Let, for gєG, gHg^-1 = {ghg^-1|hєH}.Prove that gHg^-1 is a subgroup of G.

Modern Algebra Group Theory (XLIII) Subgroups of a Group Subgroups of the type gHg^-1 Let G be a group, H a subgroup of G. Let, for gєG, gHg^-1 = {ghg^-1|hєH}. ...continues

Group Theory (XLIV): Suppose H is the only subgroup of order O(H) in the finite group G. Prove that H is a normal subgroup of G.

Modern Algebra Group Theory (XLIV) Subgroups of a Group Normal Subgroups of a Group Suppose H is the only subgroup of order O(H) in the finite group G. Prove that H is a ...continues

Normalizer of a Subgroup of a Group: If H is a subgroup of G, let N(H) = {gєG|gHg^-1 = H}. Prove that N(H) is a subgroup of G.

Modern Algebra Group Theory (XLV) 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|gH ...continues

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

Modern Algebra Group Theory (XLVI) Normal Subgroups of a Group Normalizer of a Subgroup of a Group Centralizer of a Subgroup of a Group ...continues

Group Theory (XLVII): If H is a subgroup of G, let N(H) = {gєG|gHg^-1 = H}. Prove that if H is a normal subgroup of the subgroup K in G, then K is subset N(H)( that is, N(H) is the largest subgroup of G in which H is normal).

Modern Algebra Group Theory (XLVII) Normal Subgroups of a Group Normalizer of a Subgroup of a Group Centralizer of a Subgroup of a Group If ...continues