Page 126 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 126
3.5 Nilpotent groups and p -groups
Since x = x z p , it follows that a p ≡ 1 mod p 2, hence a ≡ 1 mod p . Write a = 1 + k p. Let
l be such that k l ≡ 1 mod p . Let y = z l . Then x y = x 1+p . Since N ∩ 〈y 〉 = 1, we have
N 〈y 〉 = G .
All the groups above are clearly extraspecial.
A group G is said to be the central product of its normal subgroups G1, . . . ,Gn if G =
G1 · · ·Gn , [Gi ,G j ] = 1 for i = j , and Gi ∩ j =i G j = Z (G ).
Theorem 3.5.27 An extraspecial p -group is a central product of n nonabelian subgroups
of order p 3, and has order p 2n+1. Conversely, a finite central product of nonabelian groups
of order p 3 is an extraspecial p -group.
PROOF. Let C = Z (G ) = G , and let c be a generator of C . The group V = G /C is elemen-
tary abelian, hence a vector space over GF(p ). We have a well defined skew-symmetric
bilinear form f : V × V → GF(p ) induced by
[x , y ] = c (C x ,C y )f .
If (C x ,C y )f = 0 for all y ∈ G , then x ∈ C , thus f is nondegenerate. Thus there exists
a decomposition V = V1 ⊕ · · · ⊕ Vn where Vi is a 2-dimensional space with basis {u i , vi },
such that
(u i , vi ) f = 1,
(u i , vj ) f = 0 for i = j ,
(u i , u j ) f = 0,
(vi , vj ) f = 0.
Write u i = C xi , vi = C yi . Then Gi = 〈xi , yi 〉 is a nonabelian group of order p 3. We
have that G is the central product of G1, . . .Gn . Clearly G /C = G1/C × · · · × Gn /C , hence
|G | = p 2n+1.
Conversely, let G be the central product of G1, . . . ,Gn , where each Gi is a nonabelian
group of order p 3. Since Z (Gi ) ≤ Z (G ), it follows that Z (G ) = Z (Gi ) =∼ Cp . Beside that,
[Gi ,G j ] = 1 for i = j , and [Gi ,Gi ] = Z (Gi ) = Z (G ) for all i . Hence
[G ,G ] = [G1 · · ·Gn ,G1 · · ·Gn ] = Z (G ),
therefore G is extraspecial.
3.5.3 Enumeration of finite p -groups
It turns out that most of the finite groups are p -groups. The proof is beyond the scope
of these notes. To illustrate this result, there are 49, 910, 529, 484 different isomorphism
Since x = x z p , it follows that a p ≡ 1 mod p 2, hence a ≡ 1 mod p . Write a = 1 + k p. Let
l be such that k l ≡ 1 mod p . Let y = z l . Then x y = x 1+p . Since N ∩ 〈y 〉 = 1, we have
N 〈y 〉 = G .
All the groups above are clearly extraspecial.
A group G is said to be the central product of its normal subgroups G1, . . . ,Gn if G =
G1 · · ·Gn , [Gi ,G j ] = 1 for i = j , and Gi ∩ j =i G j = Z (G ).
Theorem 3.5.27 An extraspecial p -group is a central product of n nonabelian subgroups
of order p 3, and has order p 2n+1. Conversely, a finite central product of nonabelian groups
of order p 3 is an extraspecial p -group.
PROOF. Let C = Z (G ) = G , and let c be a generator of C . The group V = G /C is elemen-
tary abelian, hence a vector space over GF(p ). We have a well defined skew-symmetric
bilinear form f : V × V → GF(p ) induced by
[x , y ] = c (C x ,C y )f .
If (C x ,C y )f = 0 for all y ∈ G , then x ∈ C , thus f is nondegenerate. Thus there exists
a decomposition V = V1 ⊕ · · · ⊕ Vn where Vi is a 2-dimensional space with basis {u i , vi },
such that
(u i , vi ) f = 1,
(u i , vj ) f = 0 for i = j ,
(u i , u j ) f = 0,
(vi , vj ) f = 0.
Write u i = C xi , vi = C yi . Then Gi = 〈xi , yi 〉 is a nonabelian group of order p 3. We
have that G is the central product of G1, . . .Gn . Clearly G /C = G1/C × · · · × Gn /C , hence
|G | = p 2n+1.
Conversely, let G be the central product of G1, . . . ,Gn , where each Gi is a nonabelian
group of order p 3. Since Z (Gi ) ≤ Z (G ), it follows that Z (G ) = Z (Gi ) =∼ Cp . Beside that,
[Gi ,G j ] = 1 for i = j , and [Gi ,Gi ] = Z (Gi ) = Z (G ) for all i . Hence
[G ,G ] = [G1 · · ·Gn ,G1 · · ·Gn ] = Z (G ),
therefore G is extraspecial.
3.5.3 Enumeration of finite p -groups
It turns out that most of the finite groups are p -groups. The proof is beyond the scope
of these notes. To illustrate this result, there are 49, 910, 529, 484 different isomorphism
