Page 84 - 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
This result has numerous consequences for the structure of finite groups, see the
problems at the end of this section. We mention here that GAP can compute a Sylow
p -subgroup of a given group as follows:
gap> G := SymmetricGroup( 4 );;
gap> P := SylowSubgroup( G, 2 );
Group([ (1,2), (3,4), (1,3)(2,4) ])
How many Sylow 2-subgroups of S4 are there? A consequence of Sylow’s theorem is
also that if P is a Sylow p -subgroup of G , then sp = |G : NG (P)|. Thus:
gap> Index( G, Normalizer( G, P ) );
3
Thus there are three Sylow 2-subgroups of S4. All of them are conjugate to P:
gap> ConjugacyClassSubgroups( G, P );
Group( [ (1,2), (3,4), (1,3)(2,4) ] )^G
gap> Elements( last );
[Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (2,3), (1,4), (1,3)(2,4)]),
Group([ (1,3), (2,4), (1,4)(2,3) ]) ]
A finite group is said to be a p -group if every element has order a power of p . Equiv-
alently, the order of the group is p n for some n (exercise).
Proposition 3.2.25 Let G be a p -group. Then Z (G ) is non-trivial, and G contains a nor-
mal subgroup of order p .
PROOF. We may assume that G is non-abelian of order p n . Let x1, . . . , xr be the represen-
tatives of non-central conjugacy classes of G . By the Class Equation,
r
p n = |Z (G )| + |G : CG (xi )|.
i =1
Since CG (xi ) = G , the prime p divides |G : CG (xi )| for all i = 1, . . . , r . It follows that p
divides |Z (G )|. The rest is now straightforward.
Example 3.2.26 There is only one group of order p , namely Cp . Let us show that all groups
of order p 2 are abelian (hence there are only two possibilities, Cp × Cp and Cp2 ). Suppose
there exists a non-abelian group G of order p 2. Then Z (G ) =∼ Cp and G /Z (G ) =∼ Cp . Let
Z (G )x be a generator of G /Z (G ). Then G = Z (G )〈x 〉, but the latter group is abelian, which
is a contradiction.
Example 3.2.27 Let us classify all groups of order pq , where p and q are distinct primes
(for p = q see Example 3.2.26). Assume that p > q . Let P be a Sylow p -subgroup, and Q
a Sylow q -subgroup of G . Then Sylow’s theorem implies that sp = 1, i.e., P is a normal
subgroup of G . Similarly, sq ∈ {1, p }, and sp = 1 if and only if p ≡ 1 mod q . We separate
the two cases:
This result has numerous consequences for the structure of finite groups, see the
problems at the end of this section. We mention here that GAP can compute a Sylow
p -subgroup of a given group as follows:
gap> G := SymmetricGroup( 4 );;
gap> P := SylowSubgroup( G, 2 );
Group([ (1,2), (3,4), (1,3)(2,4) ])
How many Sylow 2-subgroups of S4 are there? A consequence of Sylow’s theorem is
also that if P is a Sylow p -subgroup of G , then sp = |G : NG (P)|. Thus:
gap> Index( G, Normalizer( G, P ) );
3
Thus there are three Sylow 2-subgroups of S4. All of them are conjugate to P:
gap> ConjugacyClassSubgroups( G, P );
Group( [ (1,2), (3,4), (1,3)(2,4) ] )^G
gap> Elements( last );
[Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (2,3), (1,4), (1,3)(2,4)]),
Group([ (1,3), (2,4), (1,4)(2,3) ]) ]
A finite group is said to be a p -group if every element has order a power of p . Equiv-
alently, the order of the group is p n for some n (exercise).
Proposition 3.2.25 Let G be a p -group. Then Z (G ) is non-trivial, and G contains a nor-
mal subgroup of order p .
PROOF. We may assume that G is non-abelian of order p n . Let x1, . . . , xr be the represen-
tatives of non-central conjugacy classes of G . By the Class Equation,
r
p n = |Z (G )| + |G : CG (xi )|.
i =1
Since CG (xi ) = G , the prime p divides |G : CG (xi )| for all i = 1, . . . , r . It follows that p
divides |Z (G )|. The rest is now straightforward.
Example 3.2.26 There is only one group of order p , namely Cp . Let us show that all groups
of order p 2 are abelian (hence there are only two possibilities, Cp × Cp and Cp2 ). Suppose
there exists a non-abelian group G of order p 2. Then Z (G ) =∼ Cp and G /Z (G ) =∼ Cp . Let
Z (G )x be a generator of G /Z (G ). Then G = Z (G )〈x 〉, but the latter group is abelian, which
is a contradiction.
Example 3.2.27 Let us classify all groups of order pq , where p and q are distinct primes
(for p = q see Example 3.2.26). Assume that p > q . Let P be a Sylow p -subgroup, and Q
a Sylow q -subgroup of G . Then Sylow’s theorem implies that sp = 1, i.e., P is a normal
subgroup of G . Similarly, sq ∈ {1, p }, and sp = 1 if and only if p ≡ 1 mod q . We separate
the two cases:
