Difference between revisions of "PGL(2,3) is isomorphic to S4"

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# [[uses::First isomorphism theorem]]
 
# [[uses::First isomorphism theorem]]
 
# [[uses::Equivalence of definitions of size of projective space]]
 
# [[uses::Equivalence of definitions of size of projective space]]
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# [[uses::Order formulas for linear groups of degree two]]
 
==Proof==
 
==Proof==
  
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| 3 || For <math>k</math> a finite field of size <math>q</math>, <math>\mathbb{P}^1(k)</math> has size <math>q + 1</math>. || Fact (2) || || <toggledisplay>The general formula for size of projective space of dimension <math>n</math> over a field of size <math>q</math> is <math>q^n + q^{n-1} + \dots + q + 1</math>, or <math>(q^{n+1} - 1)/(q - 1)</math>. Plug <math>n = 1</math> to get <math>q + 1</math>.</toggledisplay>
 
| 3 || For <math>k</math> a finite field of size <math>q</math>, <math>\mathbb{P}^1(k)</math> has size <math>q + 1</math>. || Fact (2) || || <toggledisplay>The general formula for size of projective space of dimension <math>n</math> over a field of size <math>q</math> is <math>q^n + q^{n-1} + \dots + q + 1</math>, or <math>(q^{n+1} - 1)/(q - 1)</math>. Plug <math>n = 1</math> to get <math>q + 1</math>.</toggledisplay>
 
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| 4 || For <math>k</math> a finite field of size <math>q</math>, <math>PGL(2,k)</math>, also denoted <math>PGL(2,q)</math> has order <math>q^3 - q</math>. || || || <toggledisplay>The order formula for <math>GL(2,k)</math> is <math>(q^2 - 1)(q^2 - q)</math> (obtained by counting the number of possible vectors in the first row times the number of possible vectors in the second row conditional to the first vector). The scalar matrices form a subgroup of size <math>q - 1</math>, so the quotient has size <math>(q^2 - 1)(q^2 - q)/(q - 1) = q(q^2 - 1) = q^3 - q</math>.</toggledisplay>
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| 4 || For <math>k</math> a finite field of size <math>q</math>, <math>PGL(2,k)</math>, also denoted <math>PGL(2,q)</math> has order <math>q^3 - q</math>. || Fact (3) || || <toggledisplay>See Fact (3) for formulas. Explicitly, the order formula for <math>GL(2,k)</math> is <math>(q^2 - 1)(q^2 - q)</math> (obtained by counting the number of possible vectors in the first row times the number of possible vectors in the second row conditional to the first vector). The scalar matrices form a subgroup of size <math>q - 1</math>, so the quotient has size <math>(q^2 - 1)(q^2 - q)/(q - 1) = q(q^2 - 1) = q^3 - q</math>.</toggledisplay>
 
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| 5 || For <math>k</math> the field of size three, we have <math>\operatorname{Sym}(\mathbb{P}^1(k)) = S_4</math> and its order is <math>24</math> and <math>|PGL(2,k)| = |PGL(2,3)| = 24</math> || || Steps (3), (4) || <toggledisplay>By Step (3), <math>\mathbb{P}^1(k)</math> has size <math>3 + 1 = 4</math>, so the symmetric group on it is <math>S_4</math>, and its order is <math>4! = 24</math>. Also, by Step (4), <math>PGL(2,3)</math> has order <math>3^3 - 3= 24</math>.</toggledisplay>
 
| 5 || For <math>k</math> the field of size three, we have <math>\operatorname{Sym}(\mathbb{P}^1(k)) = S_4</math> and its order is <math>24</math> and <math>|PGL(2,k)| = |PGL(2,3)| = 24</math> || || Steps (3), (4) || <toggledisplay>By Step (3), <math>\mathbb{P}^1(k)</math> has size <math>3 + 1 = 4</math>, so the symmetric group on it is <math>S_4</math>, and its order is <math>4! = 24</math>. Also, by Step (4), <math>PGL(2,3)</math> has order <math>3^3 - 3= 24</math>.</toggledisplay>
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{| class="sortable" border="1"
 
{| class="sortable" border="1"
! Command !! Functions used !! Output !! Meaning
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! Command !! Functions used !! Output !! Meaning !! Memory usage
 
|-
 
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| <tt>IsomorphismGroups(PGL(2,3),SymmetricGroup(4))</tt> || [[GAP:IsomorphismGroups|IsomorphismGroups]], [[GAP:PGL|PGL]], [[GAP:SymmetricGroup|SymmetricGroup]] || <tt>[ (3,4), (1,2,4) ] -> [ (1,4), (1,2,3) ]</tt> || The output is an actual mapping of generating sets for the groups (as they happen to be stored in GAP) that induces an isomorphism. The fact that a mapping is output indicates that an isomorphism ''does'' exist. If the groups were not isomorphic, an output of <tt>fail</tt> would be returned.
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| <tt>IsomorphismGroups(PGL(2,3),SymmetricGroup(4))</tt> || [[GAP:IsomorphismGroups|IsomorphismGroups]], [[GAP:PGL|PGL]], [[GAP:SymmetricGroup|SymmetricGroup]] || <tt>[ (3,4), (1,2,4) ] -> [ (1,4), (1,2,3) ]</tt> || The output is an actual mapping of generating sets for the groups (as they happen to be stored in GAP) that induces an isomorphism. The fact that a mapping is output indicates that an isomorphism ''does'' exist. If the groups were not isomorphic, an output of <tt>fail</tt> would be returned. || 22826
 
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| <tt>IdGroup(PGL(2,3)) = IdGroup(SymmetricGroup(4))</tt> || [[GAP:IdGroup|IdGroup]], [[GAP:PGL|PGL]], [[GAP:SymmetricGroup|SymmetricGroup]] || <tt>true</tt> || The ID of a group is uniquely determined by the group's isomorphism class, and non-isomorphic groups always have different IDs. This test can thus be used to check whether the groups are isomorphic.
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| <tt>IdGroup(PGL(2,3)) = IdGroup(SymmetricGroup(4))</tt> || [[GAP:IdGroup|IdGroup]], [[GAP:PGL|PGL]], [[GAP:SymmetricGroup|SymmetricGroup]] || <tt>true</tt> || The ID of a group is uniquely determined by the group's isomorphism class, and non-isomorphic groups always have different IDs. This test can thus be used to check whether the groups are isomorphic. || NA
 
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| <tt>StructureDescription(PGL(2,3))</tt> || [[GAP:StructureDescription|StructureDescription]], [[GAP:PGL|PGL]] || <tt>"S4"</tt> || The structure description of <math>PGL(2,3)</math> is "S4" meaning that it is the symmetric group of degree four.
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| <tt>StructureDescription(PGL(2,3))</tt> || [[GAP:StructureDescription|StructureDescription]], [[GAP:PGL|PGL]] || <tt>"S4"</tt> || The structure description of <math>PGL(2,3)</math> is "S4" meaning that it is the symmetric group of degree four. || NA
 
|}
 
|}

Latest revision as of 08:30, 30 January 2013

This article gives a proof/explanation of the equivalence of multiple definitions for the term symmetric group:S4
View a complete list of pages giving proofs of equivalence of definitions

Statement

The projective general linear group of degree two over field:F3 (the field of three elements) is isomorphic to symmetric group:S4.

Related facts

Similar facts

What the proof technique says for other projective general linear groups

The proof technique shows that there is an injective homomorphism from PGL(2,q) (order q^3 - q) into S_{q+1} (order (q + 1)!). In the cases q = 2 (PGL(2,2) \to S_3) and q = 3 (PGL(2,3) \to S_4) the orders on both sides are equal, hence the injective homomorphism is an isomorphism. For q \ge 4, the injective homomorphism is not an isomorphism.

Facts used

  1. First isomorphism theorem
  2. Equivalence of definitions of size of projective space
  3. Order formulas for linear groups of degree two

Proof

Step no. Assertion/construction Facts used Previous steps used Explanation
1 For any field k, GL(2,k) has a natural action on the set \mathbb{P}^1(k) of all lines through the origin in k^2. The kernel of this action is precisely the scalar matrices. [SHOW MORE]
2 The action in Step (1) descends to a faithful group action of PGL(2,k) on the set \operatorname{P}^1(k) of all lines through the origin in k^2, and hence an injective homomorphism from PGL(2,k) to the symmetric group on the set \mathbb{P}^1(k). Fact (1) Step (1) [SHOW MORE]
3 For k a finite field of size q, \mathbb{P}^1(k) has size q + 1. Fact (2) [SHOW MORE]
4 For k a finite field of size q, PGL(2,k), also denoted PGL(2,q) has order q^3 - q. Fact (3) [SHOW MORE]
5 For k the field of size three, we have \operatorname{Sym}(\mathbb{P}^1(k)) = S_4 and its order is 24 and |PGL(2,k)| = |PGL(2,3)| = 24 Steps (3), (4) [SHOW MORE]
6 For k the field of size three, the injective homomorphism of Step (2) gives an isomorphism from PGL(2,3) to S_4. Steps (2), (5) [SHOW MORE]

GAP implementation

The fact that the groups are isomorphic can be tested in any of these ways:

Command Functions used Output Meaning Memory usage
IsomorphismGroups(PGL(2,3),SymmetricGroup(4)) IsomorphismGroups, PGL, SymmetricGroup [ (3,4), (1,2,4) ] -> [ (1,4), (1,2,3) ] The output is an actual mapping of generating sets for the groups (as they happen to be stored in GAP) that induces an isomorphism. The fact that a mapping is output indicates that an isomorphism does exist. If the groups were not isomorphic, an output of fail would be returned. 22826
IdGroup(PGL(2,3)) = IdGroup(SymmetricGroup(4)) IdGroup, PGL, SymmetricGroup true The ID of a group is uniquely determined by the group's isomorphism class, and non-isomorphic groups always have different IDs. This test can thus be used to check whether the groups are isomorphic. NA
StructureDescription(PGL(2,3)) StructureDescription, PGL "S4" The structure description of PGL(2,3) is "S4" meaning that it is the symmetric group of degree four. NA