Math 5331                           Problems in Group Theory II                  

Dr. Cordero

1.     Prove that if a group G acts on a set S, , and s and t are in the same orbit, then .

2.     Let G be a group acting on itself by conjugation, i.e.  for every . For  denote the orbit of  in G by .
          (i) If   , find the orbit  for each .

              (ii) Verify that G is abelian if and only if  for each .
              (iii) Verify that in general,  if and only if center of G.

3.     Let S denote the collection of all subgroups of a finite group G. Let G act on S by conjugation, i.e.  for every . If  and , determine the orbit of H in G and the stabilizer of H in G.

4.     Prove that if a group G acts on a set S and for some  and , , then

5.     Let G be a group, H a subgroup of G, and S the set of all left cosets of H in G. Let G act on S by left multiplication, i.e.  Prove that the kernel of this action is .

6.     Let G be a finite group G which contains a subgroup  such that  does not divide .
(i). Prove that  H contains a nontrivial normal subgroup of G. (Hint: Use problem 5 above and Lagrange’s theorem.)
(ii) Prove that G is not simple. (Hint: Use Problem 6 above.)
(iii). Prove that there is no simple group of order 100. You may assume that each group of order 100 contains a subgroup of order 25.(This will be proved later.)