subgroup
Definition:
Let be a group and let be subset of . Then is a subgroup of defined under the same operation if is a group by itself (with respect to ), that is:
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is closed under the operation.
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There exists an identity element such that for all , .
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Let then there exists an inverse such that .
The subgroup is denoted likewise . We denote being a subgroup of by writing .
In addition the notion of a subgroup of a semigroup can be defined in the following manner. Let be a semigroup and be a subset of . Then is a subgroup of if is a subsemigroup of and is a group.
Properties:
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Every group is a subgroup of itself.
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The null set is never a subgroup (since the definition of group states that the set must be non-empty).
There is a very useful theorem that allows proving a given subset is a subgroup.
Theorem:
If is a nonempty subset of the group . Then is a subgroup of if and only if implies that .
Proof:
First we need to show if is a subgroup of then . Since then , because is a group by itself.
Now, suppose that if for any we have . We want to show that is a subgroup, which we will accomplish by proving it holds the group axioms.
Since by hypothesis, we conclude that the identity element is in : . (Existence of identity)
Now that we know , for all in we have that so the inverses of elements in are also in . (Existence of inverses).
Let . Then we know that by last step. Applying hypothesis shows that
so is closed under the operation.
Example:
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Consider the group . Show that is a subgroup.
The subgroup is closed under addition since the sum of even integers is even.
The identity of is also on since divides . For every there is an which is the inverse under addition and satisfies . Therefore is a subgroup of .
Another way to show is a subgroup is by using the proposition stated above. If then are even numbers and since the difference of even numbers is always an even number.
See also:
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Wikipedia, http://www.wikipedia.org/wiki/Subgroupsubgroup
Title | subgroup |
Canonical name | Subgroup |
Date of creation | 2013-03-22 12:02:10 |
Last modified on | 2013-03-22 12:02:10 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 18 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 20A05 |
Related topic | Group |
Related topic | Ring |
Related topic | FreeGroup |
Related topic | Cycle2 |
Related topic | Subring |
Related topic | GroupHomomorphism |
Related topic | QuotientGroup |
Related topic | ProperSubgroup |
Related topic | SubmonoidSubsemigroup |
Related topic | ProofThatGInGImpliesThatLangleGRangleLeG |
Related topic | AbelianGroup2 |
Related topic | EssentialSubgroup |
Related topic | PGroup4 |
Defines | trivial subgroup |