Commutative subgroup . In Algebra it is said of the algebraic structure formed by the pair <G ‘, *> that makes up a subgroup and * is a binary and commutative operation . It is also equivalent to saying abelian subgroup .
Summary
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- 1 Example 1.
- 2 Example 2.
- 3 Example 3.
- 4 Example 4.
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Definition.
- Is a set Gand the binary operation * such that <G, *> form an algebraic group is called commutative or Abelian subgroup any with that meets the fact also be a group on the same operation * and also for all x and y of G ‘it is true that x * y = y * x .
An important consequence in commutative subgroups is the following:
- The trivial subgroup <{e}, *> is always commutative.
Examples.
Example 1.
The smallest of the commutative subgroups of the group <G, *> is <{e}, *> where e is the neutral element for * in G , that is, the trivial subgroup.
Example 2.
Let be the group <{0, 1, 2, 3, 4}, + 5 > whose operation + 5 ( sum modulo 5) is defined by the table:
| + 5 | 0 | one | 2 | 3 | 4 |
| 0 | 0 | one | 2 | 3 | 4 |
| one | one | 2 | 3 | 4 | 0 |
| 2 | 2 | 3 | 4 | 0 | one |
| 3 | 3 | 4 | 0 | one | 2 |
| 4 | 4 | 0 | one | 2 | 3 |
it has no other subgroup than the trivial one, because for the rest of the subsets of G the axiom of closing groups is violated .
Example 3.
Let be the group <{e, a, b, c}, @> ( Klein’s Fourth Group ) with @ defined according to the following table:
| @ | and | to | b | c |
| and | and | to | b | c |
| to | to | and | c | b |
| b | b | c | and | to |
| c | c | b | to | and |
It has <{e}, @> , <{e, a}, @> by commutative subgroups.
Example 4.
Let be the real subset [0; 1] and the arithmetic product, these constitute a commutative subgroup of the real numbers , not the same subset and the sum, because it is not closed and does not fulfill the existence of the inverse for example for 1.
