The set of all non-zero residue classes modulo a prime number p is a group with respect to multiplication of residue classes.
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- Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.25. Prove or disprove that every group of order is abelian.
- Exercises 35. Prove that any two groups of order are isomorphic.Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.Exercises 12. Prove that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. (Hint: Consider the mapping defined by for all .)
- 7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .