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The Frattini subgroups of the dihedral and quaternion groups of order eight

Example

Let $D : = Dih (C_{4}) = D_{4}$ in the convention of $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations, so $D$ is the dihedral group of order eight, and let $Q_{8}$ be the quaternion group. Then

$Φ (D) = ⟨ r^{2} ⟩, Φ (Q_{8}) = {1, - 1} .$

For the dihedral group of order eight and $Q_{8}$ , the Frattini subgroup has order two and the Frattini quotient is $(Z / 2)^{2}$ . Both groups have generator rank two.

Facts & Assumptions

Given: The normal forms $r^{i}$ and $r^{i} s$ in $D = Dih (C_{4})$ , and the multiplication table of $Q_{8}$ ( The generalized dihedral group $Dih (A) = A ⋊ C_{2}$ for an abelian group $A$ , The quaternion group $Q_{8} = {\pm 1, \pm i, \pm j, \pm k}$ inside the nonzero quaternions).

[L1]

For every finite $2$ -group $P$ , $Φ (P) = P^{2}$ ( $Φ (P) = P^{2}$ for a finite $2$ -group).

[L2]

In $D = Dih (C_{4})$ , $r^{4} = s^{2} = 1$ , $s r s^{- 1} = r^{- 1}$ , and every element is $r^{i}$ or $r^{i} s$ ; in $Q_{8}$ , the elements $\pm i, \pm j, \pm k$ have order four and $- 1$ is the unique element of order two ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations, $Q_{8}$ is a subgroup of $H^{\times}$ with eight elements, and $- 1$ is its only element of order $2$ ).

[F1]

The commutator subgroup is generated by $[g, h] = g h g^{- 1} h^{- 1}$ (Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ ).

[F2]

The generator rank is the common size of a basis of the Frattini quotient (The generator rank $d (P)$ of a finite $p$ -group).

Verification

technique · direct

1.1givenL2F1algebra

In $D$ , $(r^{i})^{2} = r^{2 i}$ and $(r^{i} s)^{2} = 1$ , so $D^{2} = ⟨ r^{2} ⟩$ . In $Q_{8}$ , the squares are $1$ and $- 1$ , with every noncentral element squaring to $- 1$ , so $Q_{8}^{2} = {1, - 1}$ . The commutators of [F1] give the same two subgroups: $[r, s] = r (s r^{- 1} s^{- 1}) = r \cdot r = r^{2}$ and every commutator of $D$ is a power of $r^{2}$ , so $D^{'} = ⟨ r^{2} ⟩$ ; and $[i, j] = i j i^{- 1} j^{- 1} = k (- i) (- j) = k i j = k^{2} = - 1$ , so $Q_{8}^{'} = {1, - 1}$ .

2.1step 1.1L1L2F2algebra∎

Apply [L1] to step 1.1. Each quotient has order four and exponent two, with the classes of $r, s$ and of $i, j$ respectively as two-vector bases. Thus both quotients are $(Z / 2)^{2}$ , and [F2] gives generator rank two.

Depends on

Used by

The maximal subgroups of the dihedral group of order eight as Frattini hyperplanes Example

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Generating Sets, Example 6.10 (standard reference, not scraped)