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IB Mathematics HL – Option: Sets, Relations and Groups, Theory

Can anyone help me with the fundamental theorems and propositions on Group Theory??

Thanks

Can anyone help me with the fundamental theorems and propositions on Group Theory??

Thanks

- elizabeth
**Posts:**0**Joined:**Mon Jan 28, 2013 8:09 pm

IB Mathematics HL – Option: Sets, Relations and Groups, Theory

Below are listed the major Definitions, Theorems and Propositions related to Groups, Subgroups, and Cyclic Groups

1. is a group under addition modulo n. With identity 0 and the inverse of is the

2. The number of elements of a group is its order

3. The order of an element g, which is denoted by , in a group G is the smallest positive integer n such that

4. Cyclic group and g is called a generator of

5. The order of the generator is the same as the order of the group it generated.

6. Let G be a group, and let x be any element of G. Then, is a subgroup of G.

7. Let be a cyclic group of order n.

Then if and only if gcd(n,k)=1.

8. Every subgroup of a cyclic group is cyclic.

9. Every cyclic group is Abelian.

10. If G is isomorphic to H then G is Abelian if and only if H is Abelian.

11. If G is isomorphic to H then G is Cyclic if and only if H is Cyclic.

12. Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides G.

13. In a finite group, the order of each element of the group divides the prder of the group.

14. A Group of prime order is cyclic.

15. For any where e is the identity element of the group G.

16. An infinite cyclic group is isomorphic to the additive group

17. A cyclic group of order n is isomorphic to the additive group of integers modulo n.

18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.

19. Let (G, *) be a group and . If , then , where is the identity element of the group G.

20. In a Cayley table for a group (G, *), each element appears exactly once in each row and exactly once in each column.

21. A group (G,*) is called a finite group if G has only a finite number of elements. The order of the group is the number of its elements.

22. A group with an infinite number of elements is called an infinite group.

23. If (G,*) is a group, then ({e},*) and (G,*) are subgroups of (G,*) and are called trivial.

24. Let G be a group and H be a non-empty subset of G. then H is a subgroup of G if and only if for all .

25. Let G be a group and H be a finite non-empty subset of G. then H is a subgroup of G if and only if for all .

26. Let be a finite group of order n.

Then

27. Let be a finite cyclic group. Then the order of g equals the order of the group.

28. A finite group G is a cyclic group if and only if there exists an element such that the order of this element equals the order of the group ().

29. Let G be a finite cyclic group of order n. Then froe every positive divisor d of n, there exists a unique subgroup of G of order d.

30. Let G be a group of finite order n. Then the order of any element x of G divides n and

31.Let G be a group of prime order. Then g is cyclic.

Hope these help!!

Below are listed the major Definitions, Theorems and Propositions related to Groups, Subgroups, and Cyclic Groups

1. is a group under addition modulo n. With identity 0 and the inverse of is the

2. The number of elements of a group is its order

3. The order of an element g, which is denoted by , in a group G is the smallest positive integer n such that

4. Cyclic group and g is called a generator of

5. The order of the generator is the same as the order of the group it generated.

6. Let G be a group, and let x be any element of G. Then, is a subgroup of G.

7. Let be a cyclic group of order n.

Then if and only if gcd(n,k)=1.

8. Every subgroup of a cyclic group is cyclic.

9. Every cyclic group is Abelian.

10. If G is isomorphic to H then G is Abelian if and only if H is Abelian.

11. If G is isomorphic to H then G is Cyclic if and only if H is Cyclic.

12. Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides G.

13. In a finite group, the order of each element of the group divides the prder of the group.

14. A Group of prime order is cyclic.

15. For any where e is the identity element of the group G.

16. An infinite cyclic group is isomorphic to the additive group

17. A cyclic group of order n is isomorphic to the additive group of integers modulo n.

18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.

19. Let (G, *) be a group and . If , then , where is the identity element of the group G.

20. In a Cayley table for a group (G, *), each element appears exactly once in each row and exactly once in each column.

21. A group (G,*) is called a finite group if G has only a finite number of elements. The order of the group is the number of its elements.

22. A group with an infinite number of elements is called an infinite group.

23. If (G,*) is a group, then ({e},*) and (G,*) are subgroups of (G,*) and are called trivial.

24. Let G be a group and H be a non-empty subset of G. then H is a subgroup of G if and only if for all .

25. Let G be a group and H be a finite non-empty subset of G. then H is a subgroup of G if and only if for all .

26. Let be a finite group of order n.

Then

27. Let be a finite cyclic group. Then the order of g equals the order of the group.

28. A finite group G is a cyclic group if and only if there exists an element such that the order of this element equals the order of the group ().

29. Let G be a finite cyclic group of order n. Then froe every positive divisor d of n, there exists a unique subgroup of G of order d.

30. Let G be a group of finite order n. Then the order of any element x of G divides n and

31.Let G be a group of prime order. Then g is cyclic.

Hope these help!!

- miranda
**Posts:**268**Joined:**Mon Jan 28, 2013 8:03 pm

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