Subgroup

In group theory, the mathematics is a subset of a group is a non-empty subset U of G, which again is a group with respect to the link itself (that is, has all the characteristics that define a group). There is a shorthand notation U ≤ G to read as " U is subset of G". The property " associativity " carries over to every subset of G, but not necessarily the properties of " seclusion ", " existence of the neutral element " and " Existence of an inverse element ", ie, not every non-empty subset V forms of G is a subgroup. The group is called superset of the group, denoted G ≥ U.

Equivalent definitions

A non-empty subset of forms a subgroup of if and only if for any two elements in their link is in, and for each element in its inverse is also in:

Another equivalent condition: The non-empty subset of if and only a subset of, if:

From these two criteria also follows that the neutral element of in must be included.

Depending on the nature of the link, it is easier to use the first or the second criterion for the detection of the group property.

Generating a subgroup

A subset of a group generates a subgroup of. is therefore the smallest group of which contains all the elements. One can show that from the neutral element of and all links of a finite number, which themselves or their inverses are in there, :

If contains only one item, the subgroup generated writes often as you place, and it is cyclical. It contains exactly the integer powers of:

In which

The group order of the subgroup is equal to the order of the generating element.

Examples

• The whole numbers with respect to the addition of a sub-group of rational numbers.
• The quantity of permutations is a subset of the symmetric group.
• , That is a group having any combination and the amount of which consists only of the respective neutral element, any other sub-group is a group that shares this link.

Properties

From a group are always self and the one-element group subgroups. These are called the trivial subgroups of. In the case of these two sub-groups are the same and are the only subgroup corner shows other groups have at least two subgroups, namely the two mutually different trivial.

One of several sub-group is a proper subgroup called, in shorthand notation U < G.

The set of all subgroups of a group forms a complete lattice, the sub-group association. The two trivial subgroups and correspond to the zero and the unit element of the association.

Lagrange's theorem: The order ( cardinality) of each subgroup of a finite group divides the order of the group. ( The ratio of the orders is the index of the group.)

For example, if a prime number, then the cardinality of a subset of only 1 or amount. Thus, in this case, the sub-groups, the only nontrivial subgroups.

Subgroups that remain fixed under the conjugation, hot normal subgroup. With these factor groups can be formed.

Is a group sub-group which is in turn sub- group, then subgroup of. (The corresponding statement for normal subgroup does not apply. )

The intersection of arbitrary subgroups of a group is a subgroup of.