Subgroup . In the case of integers we know that they constitute an algebraic group with the usual addition. The set Z of integers has several subsets, the best known being even, odd, and prime. The objective aims to know which ones assume the algebraic group structure. Of those mentioned only the pairs with the addition; the odd ones do not, they fail in the closing property, just like the cousins. However, there are an infinity of subsets of Z that are algebraic groups, any set of all multiples of a fixed integer greater than 1 additively form a group.
In Algebra it is said of the algebraic structure formed by the pair <G ‘, *> , such that G’ is a non-empty subset of G and * is a binary operation and <G, *> is a group and it is true that <G ‘, *> is also a group.
In the case that <G ‘, *> is a subgroup and the operation is commutative within G’, one is in the presence of a commutative or abelian subgroup .
Summary
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- 1
- 2
- 1 Example 1.
- 2 Example 2.
- 3 Example 3.
- 3 Propositions
- 4 Bibliography
- 5 References
Definition.
- Let a set G beand the binary operation º an algebraic group <G, º> , any subset H, not empty, of G is called a subgroup , if H is an algebraic group with operation º of G.
An important consequence in the subgroups is the following:
- The neutral element eof G, also remains neutral element of any subgroup H of G, obviously with the inherited operation.
- {e} and G are the so-called trivial groups of G.
Examples.
Example 1.
The smallest of the 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 a <{e}, @> , <{e, a}, @> , <{e, a, b}, @> per subgroup.
Propositions
- A non-empty subset H of group G (multiplicative notation) is a subgroup of G sss
- If c and d are in H, then cd is in H
- c -1is in H whenever c is in H.
Example
If G is an algebraic group and m is a fixed element of G, then N [m] = {y in G / ym = my} is a subgroup of G, called the normalizer or centralizer of m in G. [1]
- If H and K are subgroups of the algebraic group K, then the intersection of H and K is a subgroup of G.
- If H and K are finite algebraic groups, H is a subgroup of K then o (H) / o (K),
