Math 5331 Problems in Group Theory II
1. Prove that if a group G acts on a set S, , and s and t are in the same orbit, then .
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 .
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 .
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.)