Fall ’98 Math 413 Exam 1 Prof. Brick
Do the problems in order in your bluebook.
1. Define the following terms:
(a) group (b) the order of an element
2. Given a finite abelian group G={x1,x
2,...x
n}, let w=x1·x2···xn.
Consider the claim “w=e” and the following argument:
“Given xi=e, its inverse is some xj. The group is abelian, so the product
can be rewritten so that xiand xjappear next to each other. But then
they would cancel. We can do this for each xi=e, so we eventually reach
the desired result w=e”
(a) Briefly explain what is wrong in the argument.
(b) Find a counterexample to the claim.
3. Prove that (ab)−1=b−1a−1
4. Express (2 5 7)(5143)(39721)asaproduct of disjoint cycles.
5. Prove that the intersection of two subgroups is itself a subgroup.
6. Suppose Gis a group with x2=efor all x. Prove that Gis abelian.
7. Find all of the generators of Z10.
