Modern Algebra, Group Theory (XXI): Abelian Group:Prove that a group G is abelian if every element , except the identity, is of order 2.

Modern Algebra Group Theory (XXI) Abelian Group Prove that a group G is abelian if every element , except the identity, is of order 2.

Modern Algebra, Group Theory (XXII): Group of Even Order:If G is a group of even order, prove it has an element a ≠ e satisfying a^2 = e.

Modern Algebra Group Theory (XXII) Group of Even Order If G is a group of even order, prove it has an element a ≠ e satisfying a^2 = e.

Modern Algebra, Group Theory (XXIII): Formation of a Group.

Modern Algebra Group Theory (XXIII) Formation of a Group Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an eЄG such that a.e = a for all aЄG (b) Given aЄG , there exists an element y(a)ЄG such that a.y(a) = e. Prove that G must be a group under this product.

Modern Algebra, Group Theory (XXIV): Formation of a Group.

Modern Algebra Group Theory (XXIV) Formation of a Group Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an eЄG such that a.e = a for all aЄG (b) Given aЄG , there exists an element y(a)ЄG such that a.y(a) = e. Then G must be a group under this product. Prove by an example, that the conclusion of the above problem is false if we assume instead ... click for more

Modern Algebra, Group Theory (XXV): Formation of a Group.

Modern Algebra Group Theory (XXV) Formation of a Group Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an eЄG such that e.a = a for all aЄG (b) Given aЄG , there exists an element y(a)ЄG such that y(a).a = e. Prove that G must be a group under this product.