General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.
Please note that the directions for this solved exercise differ from those for the exercises in the present assignment.
Prove the following statement: If the number of elements in a finite group with identity is even, show that there is at least one element in such that but .
Proof. Let be the number of elements of the given finite group . The assertion is that there is at least one element of other than for which , i.e., . If this were not the case then for every in one would have , i.e., and would be different elements. So the set would be the disjoint union of two element subsets of the form , and, therefore, the number of elements of would be even. Since is the disjoint union of and , and, therefore, the number of elements of would be odd. Hence, if the number of elements of is even, there must be at least one element of for which .
Read these directions carefully: for each of the following statements either provide a proof that the statement is true or label the statement as false and provide justification.
The multiplicative group of the integers mod is a cyclic group.
If is an abelian group with identity , then the set of all elements such that is a subgroup of .