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then the operation for the group, *, can be used to operate on elements, with some extra properties. Note that the o-th element of the group is equal to 1; the element o - 1 denotes the end of a cycle. For the C<sub>6</sub>case, there exists an operation x<sup>4</sup> * x<sup>2</sup> = x<sup>6</sup>, where x<sup>6</sup> would be equal to 1 in this case. | then the operation for the group, *, can be used to operate on elements, with some extra properties. Note that the o-th element of the group is equal to 1; the element o - 1 denotes the end of a cycle. For the C<sub>6</sub>case, there exists an operation x<sup>4</sup> * x<sup>2</sup> = x<sup>6</sup>, where x<sup>6</sup> would be equal to 1 in this case. | ||
| − | Cyclic groups are often represented geometrically as well. Here is one example of this: | + | Cyclic groups are often represented geometrically as well. Often times the set is depicted as a regular polygon inscribed within a circle. Here is one example of this: |
Besides the cyclic group, a symmetric group is another type of group that exists. | Besides the cyclic group, a symmetric group is another type of group that exists. | ||
| − | + | [[File:Symmetric Groups Regular Hexagon|thumbnail|Figure 2.1: Symmetric Groups Visualized]] | |
[[ Walther MA271 Fall2020 topic1|Back to Walther MA271 Fall2020 topic1]] | [[ Walther MA271 Fall2020 topic1|Back to Walther MA271 Fall2020 topic1]] | ||
Revision as of 05:12, 6 December 2020
A Note on Groups
Groups are an abstractly defined concept used in abstract algebra. An understanding of the basic concepts of groups will enhance one’s understanding of the Galois Group. A general group will first be defined, and then different types of groups will be explained.
Groups make use of an operation, which may be denoted "*". Note that this does not refer to multiplication, but an operation in the abstract.
One may define a group as a set of numbers that fulfill four conditions for an operation *:
1. The use of * between any two numbers within the set must equal another member in the set ( a * b = c)
2. The associative property exists for the operation * for all members of the set
3. There exists a member of the set i called an “identity” such that (a * i = a) or (i * a = 1)
4. Each member of the set has an invertible operation with * such that (a * b = 1) or (b * a = 1)
In addition, there are different types of groups. One such group is the cyclic group: In order to understand a cyclic group, a comparison to modulus would be appropriate. Consider a set containing the numbers 1,2,3,4,5,6. If one were to apply the modulo three to each number, the set would become (1,2,3,1,2,3). Cyclic groups work in a similar way. One may define a cyclic group using an order o by denoting Co. The order tells how many elements exist in one cycle. If one were to define a set,
C6= 1, x1, x2,x3,x4....
then the operation for the group, *, can be used to operate on elements, with some extra properties. Note that the o-th element of the group is equal to 1; the element o - 1 denotes the end of a cycle. For the C6case, there exists an operation x4 * x2 = x6, where x6 would be equal to 1 in this case.
Cyclic groups are often represented geometrically as well. Often times the set is depicted as a regular polygon inscribed within a circle. Here is one example of this:
Besides the cyclic group, a symmetric group is another type of group that exists.
