四元群

Q的環圖。每一種顏色代表連結至單位元(1)之任一元素的次方。例如，紅色的環反映了 $i^{2}=1$ 、 $i^{3}=-i$ 和 $i^{4}=1$ ，亦反映了 $(-i)^{2}=-1$ 、 $(-i)^{3}=i$ 和 $(-i)^{4}=1$ 。

在群論裡，四元群 $Q_{8}$ (Quaternion Group) 是指一個階為8的非交換群，常被簡寫為 $Q$ ，且用乘法的形式表示。包含下列8個元素：

Q=\{1,i,j,k,-1,-i,-j,-k\}

其中， $1$ 代表單位元素，且 $(-1)^{2}=1$ 。對於每個元素 $a\in Q$ ，有 $(-1)a=-a=a(-1)$ 的關係。另外，

i^{2}=j^{2}=k^{2}=ijk=-1

$Q$ 的凱萊表如下：

	1	i	j	k	−1	−i	−j	−k
1	1	i	j	k	−1	−i	−j	−k
i	i	−1	k	−j	−i	1	−k	j
j	j	−k	−1	i	−j	k	1	−i
k	k	j	−i	−1	−k	−j	i	1
−1	−1	−i	−j	−k	1	i	j	k
−i	−i	1	−k	j	i	−1	k	−j
−j	−j	k	1	−i	j	−k	−1	i
−k	−k	−j	i	1	k	j	−i	−1

需要特別留意，這個群不是交換群，例如 $ij=-ji\neq ji$ 。 $Q$ 有著漢彌爾頓群較不常見的性質：每一個 $Q$ 的子群都是其正規子群，但這個群不是交換的。每一個漢彌爾頓群都會含有一個或多個 $Q$ 。

在抽象代數裡，可以造出一個其基底為 $\{1,i,j,k\}$ 的四維實向量空間，並使用上面的乘法表和分配律來形成一個結合代數，稱為一個四元數的除環。需要注意的是，這不是在 $Q$ 上的群代數（其應該是8維的）。相反地，也可以先由四元數開始，再「定義」出由八個元素 $\{1,i,j,k,-1,-i,-j,-k\}$ 所組成之乘法子群作為四元群。

$i,j,k$ 都是 $Q$ 中階為4的元素，任意選擇其中兩個就可以生成出整個群。 $Q$ 有著下列的展現 (presentation)：

\langle x,y\mid x^{4}=1,x^{2}=y^{2},yxy^{-1}=x^{-1}\rangle

其中可以取 $i=x$ 、 $j=y$ 及 $k=xy$ 。

$Q$ 的中心及交換子群為 $\{1,-1\}$ 。其商群 $Q/\{1,-1\}$ 同構於克萊因四元群 (Klein four-group) $V$ 。而 $Q$ 的內自同構群 (Inner Automorphism Group) 同構於 $Q$ 同餘其中心，且因此也會同構於克萊因四元群。 $Q$ 的自同構群會同構於對稱群 $S_{4}$ 。 $Q$ 的外自同構群因此為 $S_{4}/V\cong S_{3}$ 。

四元群 $Q$ 亦可視為是作用於在有限體 $\mathbf {GF} (3)$ 上之二維向量空間的八個非零元素。關於其圖像，請見圖像化GL(2,p)（页面存档备份，存于互联网档案馆）。

廣義四元群

一個群若被稱為廣義四元群，則表示其有一個展現

\langle x,y\mid x^{2^{n-1}}=1,x^{2^{n-2}}=y^{2},yxy^{-1}=x^{-1}\rangle

其中 $n>3$ 為整數。這個群的階為 $2^{n}$ 。原本的四元群為 $n=3$ 時的特例。廣義四元群可以被理解為單位四元數的子群，其生成元 (generator) 為

x=e^{2\pi i/2^{n-1}}

y=j\,

廣義四元群是雙循環群此一更大類型的一類。廣義四元群有著每個交換子群都是循環群的性質。可證明一具有此性質（每個交換子群都是循環群）的有限p-群若不是循環群就是廣義四元群。

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