Math 5331 – Problems In Group Theory III

  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 .
    1. If  , find the orbit  for each .
    2. Verify that G is abelian if and only if  for each .
    3. 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 .
    1. Prove that  H contains a nontrivial normal subgroup of G. (Hint: Use problem 5 above and Lagrange’s theorem.)
    2. Prove that G is not simple. (Hint: Use Problem 6 above.)
    3. 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.)