Difference between revisions of "Subgroup"
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==Definition== | ==Definition== | ||
| − | Given a [[Group|group {{M|(G,\times_G:G\times G\rightarrow G)}} | + | Given a [[Group|group]] {{M|(G,\times_G:G\times G\rightarrow G)}} we say {{M|(H,\times_H:H\times H\rightarrow H)}} is a subgroup of {{M|(G,\times_G)}} if: |
| + | # <math>H\subset G</math> | ||
| + | # the function <math>\times_H:H\times H\rightarrow G</math> given by <math>\times_H(x,y)\mapsto\times_G(x,y)</math> has <math>\text{Range}(\times_H)\subseteq H</math> | ||
| + | #* That is to say it is closed. <math>\forall x\in H\forall y\in H[\times_H(x,y)\in H]</math> | ||
| + | # There exists an identity element <math>\in H</math>. | ||
| + | #* That is to say <math>\exists e\in H\forall x\in H[ex=xe=x]</math> where <math>xy</math> denotes <math>\times_H(x,y)</math> | ||
| + | # Every element has an inverse <math>\in H</math> | ||
| + | #* That is to say <math>\forall x\in H\exists y\in H[xy=yx=e]</math> | ||
| + | # The operation is associative | ||
| + | #* That is to say <math>\forall x\in H\forall y\in H\forall z\in H[x(yz)=(xy)z]</math> | ||
| + | |||
| + | Just like a [[Group|group]] | ||
| + | |||
| + | This makes it sound a lot harder than it really is. | ||
| + | |||
| + | ==Examples== | ||
| + | ===Even numbers=== | ||
| + | Take the group <math>(\mathbb{Z},+)</math> and define <math>H=\{z\in\mathbb{Z}|z\text{ is even}\}</math> then we have <math>H\subset\mathbb{Z}</math> and we must check it is a group. | ||
| + | |||
| + | # It is closed under <math>+</math> restricted to <math>H</math> - an even + an even = even. (proof <math>2n+2m=2(m+n)</math> and anything multiplied by 2 is even) | ||
| + | # The identity {{M|0\in H}} - so we have that. | ||
| + | # Given an <math>x\in H</math> we can see easily that the inverse, <math>-x</math> is also even and thus <math>\in H</math> | ||
| + | # Associativity is inherited | ||
| + | |||
| + | ==See also== | ||
| + | * [[Coset]] | ||
| + | * [[Normal subgroup]] | ||
| + | |||
{{Definition|Abstract Algebra}} | {{Definition|Abstract Algebra}} | ||
Latest revision as of 17:35, 15 March 2015
A subgroup (H,×H:H×H→H of a Group (G,×G:G×G→G) is a set H⊆G which is a group under the operation ×G restricted to H×H.
Contents
[hide]Definition
Given a group (G,×G:G×G→G) we say (H,×H:H×H→H) is a subgroup of (G,×G) if:
- H⊂G
- the function ×H:H×H→Ggiven by ×H(x,y)↦×G(x,y)has Range(×H)⊆H
- That is to say it is closed. ∀x∈H∀y∈H[×H(x,y)∈H]
- That is to say it is closed. ∀x∈H∀y∈H[×H(x,y)∈H]
- There exists an identity element ∈H.
- That is to say ∃e∈H∀x∈H[ex=xe=x]where xydenotes ×H(x,y)
- That is to say ∃e∈H∀x∈H[ex=xe=x]
- Every element has an inverse ∈H
- That is to say ∀x∈H∃y∈H[xy=yx=e]
- That is to say ∀x∈H∃y∈H[xy=yx=e]
- The operation is associative
- That is to say ∀x∈H∀y∈H∀z∈H[x(yz)=(xy)z]
- That is to say ∀x∈H∀y∈H∀z∈H[x(yz)=(xy)z]
Just like a group
This makes it sound a lot harder than it really is.
Examples
Even numbers
Take the group (Z,+) and define H={z∈Z|z is even} then we have H⊂Z and we must check it is a group.
- It is closed under +restricted to H- an even + an even = even. (proof 2n+2m=2(m+n)and anything multiplied by 2 is even)
- The identity 0∈H - so we have that.
- Given an x∈Hwe can see easily that the inverse, −xis also even and thus ∈H
- Associativity is inherited
