Today we are going to speak about two broad and main topics: cyclic groups and some general features of finite groups (a few additional properties and theorems).
A cyclic group is, informally speaking, a group that can be generated by a single element, the group has an element that we can call “the generator” of the group and we can “make” every element of the group by direct multiplication or “powering” the element g (or by multiplication by g once and once and again, in an additive language).
Definition (17). Cyclic group. We say a group is cyclic if
The above group is at most a group with n-elements, and x is said to be “the generator” of the cyclic group.
Propostition. If is cyclic, then G is abelian.
Check: Let with and let . Then
By the way, the product symbol/binary operation will be omited with frequency when no confusion in the composition of elements is possible.
Cyclic groups some have beautiful and interesting features:
1st. The fundamental theorem of cyclic groups says that every subgroup of a cyclic group is cyclic.
2nd. The order of any subgroup of a finite cyclic group of n elements is a divisor of n.
3rd. For every positive divisor k of n the group G has exactly one subgroup of order k.
This last property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor of , the group has at most one subgroup of order .
Proposition. Let G be a finite group, given exists .
Check: Given if there is not a such that verifies that property, then there are some numbers such that . If not, the group is NOT finite! Therefore, if , then we have
That is a contradiction and the proposition truth follows in a straightforward manner.
Proposition. Let G be a finite group. Let us define
Then, every element of is different.
Check: It there are 2 elements with then and , so n is not the minimum.
These features allow the following definition:
Definition (18). Given a finite group, and the element , the order of x is defined to be
Theorem (Lagrange). Let G be a finite group, then we have
i.e., the natural number is a divisor of , the number of elements of G.
Proposition. Let , be a subgroup of , then the number of elements of H, is a divisor of .
Some consequences of the Lagrange theorem are the following propositions:
Proposition. If is a group, and , where is a prime number, then G is a cyclic group.
Proposition. For all exists, at least, the cyclic group with elements.
We will write from now as G for shorthand notation.
Definition (19). Conjugate elements and classes of conjugacy.
Let two elements in G. They are said to be conjugated if exists an element such as
This definition provides an equivalence relation in G. The equivalence classes of conjugate elements in a group G are called “conjugacy classes” of G.
Definition (20). Conjugate elements for a subgroup. If G is a group and H is a subgroup of G, we ay that two elements are related through H, and we write if .
Proposition. Let G a group and H a subgroup of G. Then,
a) The relation of conjugation defined above is an equivalence relation.
b) The equivalence class of in this relation are in correspondence with the set , the product of elements of G by those in H.
Proposition. Let G be a group and H a subgroup. Then given an element , the set
is a subgroup of G.
Definition (21). Conjugate group. Given a subgroup of G, and an element then the subgroup
is called the conjugate subgroup of H. Alternatively, A is conjugated with H if and only if (iff)
See you in the next group theory blog post.