Page 125 - 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. 125
mož Moravec: Some Topics in the Theory of Finite Groups 113
PROOF. Let M be a maximal subgroup of G . Then M G and |G : M | = p . It follows that
γ2(G )G p ≤ M , hence γ2(G )G p ≤ Frat(G ). On the other hand, G /γ2(G )G p is an elementary
abelian p -group, hence Frat(G /γ2(G )G p ) = 1. It follows that Frat(G ) ≤ γ2(G )G p .
Let G = 〈x1, . . . , xs 〉 and F = Frat(G ). Then G = G /F = 〈F x1, . . . , F xs 〉. The group G
is a vector space over GF(p ), hence it has a basis {F xi1 , . . . , F xir }. Write Y = 〈xi1 , . . . , xir 〉.
Then G = 〈Y, F 〉, hence G = 〈Y 〉.
Let G be a finite p -group. By the Burnside Basis Theorem, we can think of G / Frat(G )
as a vector space over GF(p ).
Corollary 3.5.25 Let G be a finite p -group and d the minimal number of generators of G .
Then d = dimG F (p) G / Frat(G ).
Extraspecial p -groups
A finite p -group is said to be extraspecial if G = Z (G ) =∼ Cp .
Proposition 3.5.26 Let G be a nonabelian group of order p 3. If p is odd, then G is isomor-
phic with
〈x , y | x p = y p = 1, [x , y ]x = [x , y ]y = [x , y ]〉
or
〈x , y | x p2 = 1 = y p , x y = x 1+p 〉.
These groups have exponent p and p 2 respectively. If p = 2, then G is isomorphic with D8
or quaternion group Q8. In particular, all non-abelian groups of order p 3 are extraspecial.
PROOF. All the groups given above have order p 3. For p = 2, the assertion follows from
the description of all groups of order 8 (exercise).
Assume that p is odd. We consider two cases:
Case 1. All elements of G have order p . Let z ∈ Z (G )\{1} and let x ∈/ Z (G ). Then
〈z , x 〉 = 〈z 〉 × 〈x 〉 is a subgroup of order p 2, hence it is a maximal subgroup and thus
normal in G . Choose w ∈/ 〈z , x 〉. Then G = 〈z , x , w 〉. We have that x w = x a z b for some
0 ≤ a ,b < p . If a = 0, then x y ∈ Z (G ), hence x ∈ Z (G ), a contradiction. Thus there exists
c such that a c ≡ 1 mod p . Let t = w c . We have that G = 〈z , x , t 〉, and x t = x z b for some
0 ≤ b < p . As G is nonabelian, b = 0, hence there exists d such that b d ≡ 1 mod p . Put
y = t d . Then we get [x , y ] = z and G = 〈x , y 〉. We have
x p = y p = 1, [x , y ]x = [x , y ]y = [x , y ],
as required.
Case 2. G contains an element x of order p 2. Let N = 〈x 〉. As N is a maximal subgroup
of G , N is normal in G . Choose z ∈ G \N of order p . There exists a ∈ such that x z = x a .
PROOF. Let M be a maximal subgroup of G . Then M G and |G : M | = p . It follows that
γ2(G )G p ≤ M , hence γ2(G )G p ≤ Frat(G ). On the other hand, G /γ2(G )G p is an elementary
abelian p -group, hence Frat(G /γ2(G )G p ) = 1. It follows that Frat(G ) ≤ γ2(G )G p .
Let G = 〈x1, . . . , xs 〉 and F = Frat(G ). Then G = G /F = 〈F x1, . . . , F xs 〉. The group G
is a vector space over GF(p ), hence it has a basis {F xi1 , . . . , F xir }. Write Y = 〈xi1 , . . . , xir 〉.
Then G = 〈Y, F 〉, hence G = 〈Y 〉.
Let G be a finite p -group. By the Burnside Basis Theorem, we can think of G / Frat(G )
as a vector space over GF(p ).
Corollary 3.5.25 Let G be a finite p -group and d the minimal number of generators of G .
Then d = dimG F (p) G / Frat(G ).
Extraspecial p -groups
A finite p -group is said to be extraspecial if G = Z (G ) =∼ Cp .
Proposition 3.5.26 Let G be a nonabelian group of order p 3. If p is odd, then G is isomor-
phic with
〈x , y | x p = y p = 1, [x , y ]x = [x , y ]y = [x , y ]〉
or
〈x , y | x p2 = 1 = y p , x y = x 1+p 〉.
These groups have exponent p and p 2 respectively. If p = 2, then G is isomorphic with D8
or quaternion group Q8. In particular, all non-abelian groups of order p 3 are extraspecial.
PROOF. All the groups given above have order p 3. For p = 2, the assertion follows from
the description of all groups of order 8 (exercise).
Assume that p is odd. We consider two cases:
Case 1. All elements of G have order p . Let z ∈ Z (G )\{1} and let x ∈/ Z (G ). Then
〈z , x 〉 = 〈z 〉 × 〈x 〉 is a subgroup of order p 2, hence it is a maximal subgroup and thus
normal in G . Choose w ∈/ 〈z , x 〉. Then G = 〈z , x , w 〉. We have that x w = x a z b for some
0 ≤ a ,b < p . If a = 0, then x y ∈ Z (G ), hence x ∈ Z (G ), a contradiction. Thus there exists
c such that a c ≡ 1 mod p . Let t = w c . We have that G = 〈z , x , t 〉, and x t = x z b for some
0 ≤ b < p . As G is nonabelian, b = 0, hence there exists d such that b d ≡ 1 mod p . Put
y = t d . Then we get [x , y ] = z and G = 〈x , y 〉. We have
x p = y p = 1, [x , y ]x = [x , y ]y = [x , y ],
as required.
Case 2. G contains an element x of order p 2. Let N = 〈x 〉. As N is a maximal subgroup
of G , N is normal in G . Choose z ∈ G \N of order p . There exists a ∈ such that x z = x a .
