3. Let G be a group of order 120, H a subgroup of
order 24 such that there exists a left coset of H (different
from H itself) which is equal to a right coset of H. Prove that
H is normal in G.
Solution

4. Show that no group of order 192 is simple.
Solution

5. Let G be a finite group and H be a proper subgroup
of G. Show that there is an element of G that is not in any
conjugate of H. Prove that this result does not hold if G is
infinite.
Solution

6. Prove that the groups (R,+) and
(R+*,x) are isomorphic. Show that
G=(Q+*,x) is a free abelian group, and that
there exist infinitely many homomorphisms from G to
H=(Q,+). However, show that there exists only one
homomorphism from H to G.
Solution

7. If G is a group of odd order, show that whenever x
is not the neuter element, x and x^(-1) are not conjugated.
Solution

8. Let G be a finite p-group with a unique subgroup of index p. Show that G is cyclic.
Solution

9. Assume G is a finite group and H is a normal subgroup of G. P is a p-Sylow subgroup of
H, and N=N_G(P). Show that G = NH.
Solution

10. Show that every group G of order 992 is solvable.
Solution