## Exercises

1. If , , a group, and . Show that the double coset is such that where ranges over certain elements of .

2. Let , , a group, and . Let be defined by , where and . Show is well-defined, 1-1, and onto.

3. Following the proof of part (b) of Theorem 8.2.2, prove part (a), i.e., the number of right cosets of in is .

4. Find the double coset decomposition of with respect to .

5. Let be a finite group and such that , and any two distinct conjugate subgroups of have only the identity element in common. Let be the set of elements of not contained in nor in any of its conjugates, together with the identity. Show that .

HINT: First use the given together with Lagrange's Theorem (in particular equation (4.8)) and Theorem 6.1.1 to show that . Next decompose into double cosets with respect to and and use equation (8.6). Now the identity belongs to some double coset, so we may assume that , in the line before equation (8.6). Finally this implies that in (8.6) , but all the other . (Why?) Use the resulting relation to get the desired result.

David Joyner 2007-08-06