subgroup
Definition:
Let (G,*) be a group and let K be subset of G. Then K is a subgroup of G defined under the same operation
if K is a group by itself (with respect to *), that is:
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K is closed under the * operation.
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There exists an identity element
e∈K such that for all k∈K, k*e=k=e*k.
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Let k∈K then there exists an inverse
k-1∈K such that k-1*k=e=k*k-1.
The subgroup is denoted likewise (K,*). We denote K being a subgroup of G by writing K≤G.
In addition the notion of a subgroup of a semigroup can be defined in the following manner. Let (S,*) be a semigroup and H be a subset of S. Then H is a subgroup of S if H is a subsemigroup of S and H 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 K is a nonempty subset of the group G. Then K is a subgroup of G if and only if s,t∈K implies that st-1∈K.
Proof:
First we need to show if K is a subgroup of G then st-1∈K. Since s,t∈K then st-1∈K, because K is a group by itself.
Now, suppose that if for any s,t∈K⊆G we have st-1∈K. We want to show that K is a subgroup, which we will accomplish by proving it holds the group axioms.
Since tt-1∈K by hypothesis, we conclude that the identity element is in K: e∈K. (Existence of identity)
Now that we know e∈K, for all t in K we have that et-1=t-1∈K so the inverses of elements in K are also in K. (Existence of inverses).
Let s,t∈K. Then we know that t-1∈K by last step. Applying hypothesis shows that
s(t-1)-1=st∈K |
so K is closed under the operation. QED
Example:
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Consider the group (ℤ,+). Show that(2ℤ,+) is a subgroup.
The subgroup is closed under addition since the sum of even integers is even.
The identity 0 of ℤ is also on 2ℤ since 2 divides 0. For every k∈2ℤ there is an -k∈2ℤ which is the inverse under addition and satisfies -k+k=0=k+(-k). Therefore (2ℤ,+) is a subgroup of (ℤ,+).
Another way to show (2ℤ,+) is a subgroup is by using the proposition
stated above. If s,t∈2ℤ then s,t are even numbers and s-t∈2ℤ 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 |