1. Show that Q/Z does not have any proper
subgroup with finite index.
2. Let G be a finite group and a,b two elements of
order p prime, and b does not belong to .
Show that G contains at least
p^2-1 elements of order p.
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.
4. Show that no group of order 192 is simple.
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
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.
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.
8. Let G be a finite p-group with a unique subgroup of index p. Show that G is cyclic.
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.
10. Show that every group G of order 992 is solvable.
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