The Order of an Element of a group: If aєG, a^m = e prove that O(a)|m.

Modern Algebra Group Theory (XXXII) Subgroups of a Group The Order of an Element of a group If aєG, a^m = e prove that O(a)|m.

The Order of an Element of a group: If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find O(b).

Modern Algebra Group Theory (XXXIII) Subgroups of a Group The Order of an Element of a group If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find O(b).

Cosets of Subgroups of a Group and Normal Subgroups of a Group: If H is a subgroup of G such that the product of two right cosets of H in G is again a right coset of H in G, prove that H is normal in G.

Modern Algebra Group Theory (XXXIV) Cosets of Subgroups of a Group Normal Subgroups of a Group If H is a subgroup of G such that the product of two right cosets of H in G is again a right coset of H in G, prove that H is normal in G.

Group Theory (XXXV): A subgroup N of G is a normal subgroup of G if and only if the product of two right cosets of N in G is again a right coset of N in G.

Modern Algebra Group Theory (XXXV) Cosets of Subgroups of a Group Normal Subgroups of a Group A subgroup N of G is a normal subgroup of G if and only if the product of two right cosets of N in G is again a right coset of N in G.

Group Theory (XXXVI): If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.

Modern Algebra Group Theory (XXXVI) Index of a Subgroup of a Group and Normal Subgroups of a Group If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.