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Write out the Cayley table for Sym(3), Dih(8), and the quaternion group


 

Write out the group tables for $S_3$, $D_8$, and $Q_8$.


Solution:

Group table for $S_3$:

1(1 2)(1 3)(2 3)(1 2 3)(1 3 2)
11(1 2)(1 3)(2 3)(1 2 3)(1 3 2)
(1 2)(1 2)1(1 2 3)(1 3 2)(1 3)(2 3)
(1 3)(1 3)(1 3 2)1(1 2 3)(2 3)(1 2)
(2 3)(2 3)(1 2 3)(1 3 2)1(1 2)(1 3)
(1 2 3)(1 2 3)(2 3)(1 2)(1 3)(1 3 2)1
(1 3 2)(1 3 2)(1 3)(2 3)(1 2)1(1 2 3)

Group table for $D_8$:

$1$$r$$r^2$$r^3$$s$$sr$$sr^2$$sr^3$
$1$$1$$r$$r^2$$r^3$$s$$sr$$sr^2$$sr^3$
$r$$r$$r^2$$r^3$$1$$sr^3$$s$$sr$$sr^2$
$r^2$$r^2$$r^3$$1$$r$$sr^2$$sr^3$$s$$sr$
$r^3$$r^3$$1$$r$$r^2$$sr$$sr^2$$sr^3$$s$
$s$$s$$sr$$sr^2$$sr^3$$1$$r$$r^2$$r^3$
$sr$$sr$$sr^2$$sr^3$$s$$r^3$$1$$r$$r^2$
$sr^2$$sr^2$$sr^3$$s$$sr$$r^2$$r^3$$1$$r$
$sr^3$$sr^3$$s$$sr$$sr^2$$r$$r^2$$r^3$$1$

Group table for $Q_8$:

$1$$-1$$i$$-i$$j$$-j$$k$$-k$
$1$$1$$-1$$i$$-i$$j$$-j$$k$$-k$
$-1$$-1$$1$$-i$$i$$-j$$j$$-k$$k$
$i$$i$$-i$$-1$$1$$k$$-k$$-j$$j$
$-i$$-i$$i$$1$$-1$$-k$$k$$j$$-j$
$j$$j$$-j$$-k$$k$$-1$$1$$i$$-i$
$-j$$-j$$j$$k$$-k$$1$$-1$$-i$$i$
$k$$k$$-k$$j$$-j$$-i$$i$$-1$$1$
$-k$$-k$$k$$-j$$j$$i$$-i$$1$$-1$

Linearity

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