The fundamental theorems and propositions on Group Theory from www.ibmaths4u.com
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.
Re: IB Maths HL Option: Sets, Relations and Groups
1.
2. The number of elements of a group is its order
3. The order of an element g, which is denoted by
4. Cyclic group
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,
7. Let
Then
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
16. An infinite cyclic group is isomorphic to the additive group
17. A cyclic group of order n is isomorphic to the additive group
18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.
19. Let (G, *) be a group and
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
Then
27. Let
28. A finite group G is a cyclic group if and only if there exists an element
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.
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