Does the converse of Lagrange's Theorem hold correct for a cyclic group? Either give a proof or write a counter example.
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- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
- Label each of the following statements as either true or false. The Cayley table for a group will always be symmetric with respect to the diagonal from upper left to lower right.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.
