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# Subgroup

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(See attached file for full problem description)

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State whether the following are true or false with reasons:

1. If a in S6, then an =1 for some n greater than or equal to 1.

2. If axa-1=bxb-1, then a=b

3. The function e:N x N&#61664;N, defined by e(m,n)=mn is an associative operation.

4. Every infinite group contains an element of infinite order.

5. Let G be a finite group in which every element has a square root; that is, for each x in G, there exists y in G with y=x2. Prove that every element in G has a unique square root.

6. If H is a subgroup of K and K is a subgroup of G, then H is a subgroup of G.

7. If H is a subgroup of G, then the intersection of two left cosets of H is a left coset of H.

8. The intersection of two cyclic subgroups of G is a cyclic subgroup.

9. If X is a finite subset of G, then <X> is a finite subgroup.

10. If X is an infinite set, then
F={a in SX: a moves only finitely many elements of X} is a subgroup of SX

11. Every proper subgroup of S3 is cyclic.

12. Every proper subgroup of S4 is cyclic.

13. If H and K are subgroups of a group G and if the orders of H and K are relatively prime, prove that H intersects K={1}.

14. Prove that every infinite group contains infinitely many subgroups.

15. If p is a prime, any two groups of order p are isomorphic.

16. The subgroup <(1 2)> is a normal subgroup of S3

17. The subgroup <(1 2 3)> is a normal subgroup of S3

18. If G is a group, then Z(G)=G if and only if G is abelian.

19. The 3-cycles (7 6 5) and (5 26 34) are conjugate in S100

20. If every subgroup of a group G is a normal subgroup, then G is abelian.

21. Prove that a finite p-group G is simple if and only if the order of G=p.

22. If a group G acts on a set X and if x,y in X, then Gx is isomorphic to Gy

23. If a group G acts on a set X, and if x,y in X lie in the same orbit, then Gx is isomorphic to Gy
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