State the given equational problems with true or false

Question

Questions:

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→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 is a finite subgroup.

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

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 G_x is isomorphic to G_y

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