Math Problems: Group Theory

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Problem 1

1. Show that Q/Z does not have any proper subgroup with finite index.
Solution

Problem 2

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.
Solution

Problem 3

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

Problem 4

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

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10