Page 76 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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3.2 Basic notions and examples
Example 3.2.8 Let G = 〈x , y | x 2 = y 3 = (x y )2 = 1〉. We claim that G ∼= S3. Denote a = (1 2)
and b = (1 2 3). Then a 2 = b 3 = (a b )2 = 1. By von Dyck’s Lemma, there exists a surjective
homomorphism φ : G → 〈a ,b 〉 = S3. Now consider G . We have that y x = x y 2, hence every
element of G can be written as x m y n , where 0 ≤ m ≤ 1, 0 ≤ n ≤ 2. It follows that |G | ≤ 6.
Comparing the orders, we conclude that φ must be an isomorphism between G and S3.
Another proof can be done with GAP:
gap> F := FreeGroup("x", "y");;
gap> AssignGeneratorVariables(F);;
#I Assigned the global variables [ x, y ]
gap> G := F / [x^2, y^3, (x*y)^2];;
gap> StructureDescription(G);
"S3"
In general, the group Sn has a following presentation:
〈x1, . . . , xn−1 | x 2 = 1, [xi , x j ] = 1, xi xi +1xi = xi +1xi xi +1 for all i and j = i ± 1〉.
i
Here xi corresponds to the transposition (i i + 1). This is left as an exercise.
Using GAP, one can also construct subgroups generated by certain sets of elements,
and normal closures of subgroups. It is also possible to test memberships to subgroups.
gap> G := SymmetricGroup( 5 );
Sym( [ 1 .. 5 ] )
gap> H := Subgroup( G, [(1, 2), (1, 3)]);
Group([ (1,2), (1,3) ])
gap> Order( H );
6
gap> (1,2,3,4) in H;
false
gap> N := NormalClosure(G, H);
Group([ (2,3), (1,3,2), (2,4), (3,5) ])
gap> Order( N );
120
gap> StructureDescription( H );
"S3"
gap> StructureDescription( N );
"S5"
The parity of a permutation g ∈ Sn is defined to be the parity of the number n − c (g ),
where c (g ) is the number of cycles of g (including the cycles of length 1). We regard the
parity as an element of 2. One can show that the parity is a homomorphism from Sn
onto the group 2. Its kernel consists of all permutations of even parity. It is denoted by
An and called the alternating group on n letters.
Alternating groups can be constructed with GAP:
Example 3.2.8 Let G = 〈x , y | x 2 = y 3 = (x y )2 = 1〉. We claim that G ∼= S3. Denote a = (1 2)
and b = (1 2 3). Then a 2 = b 3 = (a b )2 = 1. By von Dyck’s Lemma, there exists a surjective
homomorphism φ : G → 〈a ,b 〉 = S3. Now consider G . We have that y x = x y 2, hence every
element of G can be written as x m y n , where 0 ≤ m ≤ 1, 0 ≤ n ≤ 2. It follows that |G | ≤ 6.
Comparing the orders, we conclude that φ must be an isomorphism between G and S3.
Another proof can be done with GAP:
gap> F := FreeGroup("x", "y");;
gap> AssignGeneratorVariables(F);;
#I Assigned the global variables [ x, y ]
gap> G := F / [x^2, y^3, (x*y)^2];;
gap> StructureDescription(G);
"S3"
In general, the group Sn has a following presentation:
〈x1, . . . , xn−1 | x 2 = 1, [xi , x j ] = 1, xi xi +1xi = xi +1xi xi +1 for all i and j = i ± 1〉.
i
Here xi corresponds to the transposition (i i + 1). This is left as an exercise.
Using GAP, one can also construct subgroups generated by certain sets of elements,
and normal closures of subgroups. It is also possible to test memberships to subgroups.
gap> G := SymmetricGroup( 5 );
Sym( [ 1 .. 5 ] )
gap> H := Subgroup( G, [(1, 2), (1, 3)]);
Group([ (1,2), (1,3) ])
gap> Order( H );
6
gap> (1,2,3,4) in H;
false
gap> N := NormalClosure(G, H);
Group([ (2,3), (1,3,2), (2,4), (3,5) ])
gap> Order( N );
120
gap> StructureDescription( H );
"S3"
gap> StructureDescription( N );
"S5"
The parity of a permutation g ∈ Sn is defined to be the parity of the number n − c (g ),
where c (g ) is the number of cycles of g (including the cycles of length 1). We regard the
parity as an element of 2. One can show that the parity is a homomorphism from Sn
onto the group 2. Its kernel consists of all permutations of even parity. It is denoted by
An and called the alternating group on n letters.
Alternating groups can be constructed with GAP:
