Modern Algebra, Group Theory (VI): To determine whether the system described is a group. G = set of all rational numbers with odd denominators, a.b ≡ a + b, the usual addition of rational numbers.

Modern Algebra Group Theory (VI) To determine whether the system described is a group. G = set of all rational numbers with odd denominators, a.b ≡ a + b, the usual addition of rational numbers.

Modern Algebra, Group Theory (VII):To prove that if G is an abelian group, then for all a,bЄG and all integers n, (a.b)^n=a^n.b^n.

Modern Algebra Group Theory (VII) To prove that if G is an abelian group, then for all a,bЄG and all integers n, (a.b)^n=a^n.b^n.

Modern Algebra, Group Theory (VIII): If is a group such that (ab)^2=a^2b^2 for all a,bЄG, show that G must be abelian. Or, Show that the group G is abelian iff (ab)^2=a^2b^2.

Modern Algebra Group Theory (VIII) If G is a group such that (ab)^2=a^2b^2 for all a,bЄG, show that G must be abelian. Or, Show that the group G is abelian iff (ab)^2=a^2b^2.

Modern Algebra, Group Theory (IX): If G is a group in which (a.b)^i =a^i.b^i for three consecutive integers i for all a,bЄG , show that G is abelian.

Modern Algebra Group Theory (IX) If G is a group in which (a.b)^i =a^i.b^i for three consecutive integers i for all a,bЄG , show that G is abelian.

Modern Algebra, Group Theory (X): In a group G in which (a.b)^i =a^i.b^i for three consecutive integers for all a,bЄG , then G is abelian.Show that the conclusion does not follow if we assume the relation (a.b)^i =a^i.b^i for just two consecutive integers.

Modern Algebra Group Theory (X) In a group G in which (a.b)^i =a^i.b^i for three consecutive integers for all a,bЄG , then G is abelian. Show that the conclusion does not follow if we assume the relation (a.b)^i =a^i.b^i for just two consecutive integers.