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.)