PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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$Dih (C_{4})$ and $Q_{8}$ are extraspecial of order $8$ , with six and two solutions of $x^{2} = 1$ respectively

Statement

Both $Dih (C_{4})$ and $Q_{8}$ are extraspecial groups of order $8$ : each is nonabelian, each has centre equal to its derived subgroup of order two, and each has elementary abelian central quotient. In $Dih (C_{4})$ there are exactly six solutions of $x^{2} = 1$ , and in $Q_{8}$ exactly two.

Facts & Assumptions

Given: The generalized dihedral group $Dih (C_{4}) = ⟨ r ⟩ ⋊ ⟨ s ⟩$ with $r$ of order four, and the quaternion group $Q_{8} = {1, - 1, i, - i, j, - j, k, - k}$ .

[F1]

$Z (G) : = {z \in G : z g = g z for every g \in G}$ (The center $Z (G)$ of a group).

[F2]

For $g, h \in G$ the commutator is $[g, h] : = g h g^{- 1} h^{- 1}$ , and $[G, G]$ is the subgroup generated by all commutators (Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ ).

[F3]

The generalized dihedral group of an abelian group $A$ is $Dih (A) = A ⋊ C_{2}$ with the nonidentity element of $C_{2}$ acting by inversion ( The generalized dihedral group $Dih (A) = A ⋊ C_{2}$ for an abelian group $A$ ).

[F4]

$Q_{8} : = {1, - 1, i, - i, j, - j, k, - k}$ inside the nonzero quaternions, where $i^{2} = j^{2} = k^{2} = - 1$ , $i j = k$ , $j k = i$ , $k i = j$ , $j i = - k$ , $k j = - i$ and $i k = - j$ (The quaternion group $Q_{8} = {\pm 1, \pm i, \pm j, \pm k}$ inside the nonzero quaternions, The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ ).

[L1]

For $n \geq 1$ , $Dih (C_{n}) = C_{n} ⋊ C_{2}$ has order $2 n$ , and with $C_{n} = ⟨ r ⟩$ and $C_{2} = ⟨ s ⟩$ one has $r^{n} = s^{2} = 1$ , $s r s^{- 1} = r^{- 1}$ , and every element has a unique form $r^{i}$ or $r^{i} s$ with $0 \leq i < n$ ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L2]

$Q_{8}$ is a subgroup of the nonzero quaternions with $∣ Q_{8} ∣ = 8$ ; the element $1$ is the only element of order $1$ , $- 1$ is the only element of order $2$ , and each of $\pm i, \pm j, \pm k$ has order $4$ ( $Q_{8}$ is a subgroup of $H^{\times}$ with eight elements, and $- 1$ is its only element of order $2$ ).

[L3]

For a finite $p$ -group $P$ the following are equivalent: $P$ is extraspecial; $P$ is nonabelian, $∣ Z (P) ∣ = p$ and $P / Z (P)$ is elementary abelian; $P$ is nonabelian and $Z (P) = P^{'} = Φ (P)$ has order $p$ (Three equivalent descriptions of an extraspecial $p$ -group).

[L4]

For a finite group $G$ and $H \leq G$ , $∣ G ∣ = [G : H] ∣ H ∣$ (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

[L5]

If the quotient group $G / Z (G)$ is cyclic, then $G$ is abelian (If $G / Z (G)$ is cyclic, then $G$ is abelian).

[L6]

For $N ⊴ G$ , the quotient $G / N$ is abelian if and only if $[G, G] \subseteq N$ ( $G / N$ is abelian if and only if $[G, G] \subseteq N$ ).

[L7]

An elementary abelian $p$ -group is a finite abelian $p$ -group in which every nonidentity element has order $p$ (Elementary abelian $p$ -groups).

[L8]

$⟨ S ⟩$ is the smallest subgroup of $G$ containing $S$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[L9]

$ord (g)$ is the least $k \geq 1$ with $g^{k} = e$ (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity).

Proof

technique · direct

1.1

In $D = Dih (C_{4})$ the relation $s r s^{- 1} = r^{- 1}$ gives $s r^{- 1} = r s$ , so $r (r^{i} s) r^{- 1} = r^{i + 1} s r^{- 1} = r^{i + 2} s$ , which differs from $r^{i} s$ because $r^{2} \neq 1$ ; and $r^{i}$ is central exactly when $r^{- i} = r^{i}$ , that is when $r^{2 i} = 1$ , that is when $i$ is $0$ or $2$ . So $Z (D) = {1, r^{2}}$ , of order two, and $D$ is nonabelian of order eight.

F1F3L1

1.2

In $D$ one has $[s, r] = s r s^{- 1} r^{- 1} = r^{- 2} = r^{2}$ ; modulo $⟨ r^{2} ⟩$ the images of $r$ and $s$ commute and generate, so $[D, D] \leq ⟨ r^{2} ⟩$ and therefore $[D, D] = ⟨ r^{2} ⟩ = Z (D)$ .

F2L1L6L8

1.3

In $D$ the squares are $1^{2} = 1$ , $(r^{2})^{2} = 1$ , $(r^{i} s)^{2} = r^{i} (s r^{i} s^{- 1}) s^{2} = r^{i} r^{- i} = 1$ for each of the four values of $i$ , while $r$ and $r^{3}$ have order four. So exactly six elements satisfy $x^{2} = 1$ .

L1L9

1.4

In $Q_{8}$ the element $- 1$ is central, and $i$ is not, since $i j = k$ while $j i = - k$ and these are distinct elements of the eight-element set; the same computation excludes $\pm i, \pm j, \pm k$ , since $- x$ is central exactly when $x$ is. So $Z (Q_{8}) = {1, - 1}$ , of order two, and $Q_{8}$ is nonabelian of order eight.

F1F4L2

1.5

In $Q_{8}$ one has $[i, j] = i j i^{- 1} j^{- 1} = k (- i) (- j) = (- j) (- j) = j^{2} = - 1$ ; modulo ${1, - 1}$ the images of $i$ and $j$ commute and generate, so $[Q_{8}, Q_{8}] = {1, - 1} = Z (Q_{8})$ .

F2F4L2L6L8

1.6

In $Q_{8}$ the only solutions of $x^{2} = 1$ are $1$ and $- 1$ , because every other element has order four.

L2L9

2.1

For each of the two groups the central quotient has order four by Lagrange, is abelian because the derived subgroup equals the centre, and is not cyclic, since a cyclic central quotient would force commutativity; an abelian group of order four that is not cyclic has all nonidentity elements of order two, hence is elementary abelian.

L4L5L6L7step 1.1step 1.2step 1.4step 1.5

3.1

Each group is therefore a nonabelian group of order $2^{3}$ with centre of order two and elementary abelian central quotient, so the second description in the characterisation makes both extraspecial; the solution counts are those of steps 1.3 and 1.6.

L3step 1.1step 1.3step 1.4step 1.6step 2.1∎

Remarks

The two solution counts are what separate the two groups: an isomorphism would carry solutions of $x^{2} = 1$ to solutions of $x^{2} = 1$ , and six is not two. Nothing about the centres or the derived subgroups distinguishes them, since those agree.

Both sources write the dihedral group of order eight as $D_{8}$ ; this library writes $D_{n}$ for the dihedral group of order $2 n$ , so the group here is $Dih (C_{4})$ , which is $D_{4}$ in that notation.

Depends on

Used by

Dependency tree · two levels

63 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

D. A. Craven, The Theory of p-Groups, Example 3.2 (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, Exercise 2.33 (standard reference, not scraped)

Dih⁡(C4)Dih(C4​) and Q8Q8​ are extraspecial of order 88, with six and two solutions of x2=1x2=1 respectively