Group Theory Homework 3 --- Due Oct 27

For all true/false questions, you must prove your answer. Hint: A good way to start many true/false questions is to find some examples.

1. Suppose g is an element of order k in a group G. (o(g)=k). Prove that g^-1 is of order k.

Hint for 2 and 3. One of these is true and one is false. Why are these two situations different? Can you generalize either 2 or 3?

2. True/false. In any finite abelian group G, the product of all elements of order 2 is the identity.

3. True/false. In any finite abelian group G, the product of all elements of order 3 is the identity.

4. True/false. If o(a)=o(b)=2, then ab=ba.

5. Calculate (1,2,3)(4,3)(1,2,3)^-1 Write youranswer in both cycle form and two-row form.

6. Is the following a group? S={(x,y)|x and y are integers} where (a,b)·(c,d) = (a+c+1,b+d+1). Why or why not?