I am working through Maxfield and Maxfield's Abstract Algebra and Solution by Radicals, and the following exercise is included after discussing permutations and Cayley's Theorem:
Two rows of a certain Cayley table are
r q s 1 u v t w
q s 1 u v t w r
he rows show the eight distinct group members. The group identity is "1." Deduce the rest of the table.
At first, I assumed that the table was written in 1qrstuvw order, but that cannot be true, as it would cause w to occur twice in one column of the Cayley table.
Next, I tried assuming that the table was in abcdefgh order and finding the table by brute force, but I couldn't find out how to make that work beyond identifying inverses.
Finally, I tried to use the isomorphism between the permutation group and the relevant subgroup of $S_8$ used in the proof of Cayley's Theorem, but I couldn't figure out how to get started, as I don't know the order of the elements in the Cayley table.
Thanks so much!!
Best Answer
Each group $G$ is isomorphic to a group $H$ of permutations via Cayley's Theorem.
The elements of $H$ are given by permuting those of $G$ according to the rows (or columns) of the Cayley table with respect to left (or right, respectively) multiplication by elements of $G$. See the standard proof of Cayley's Theorem for details.
It follows from the row
$$\hat\sigma=(r,q, s, 1, u, v, t, w)$$
and
$$\hat\tau=(q, s, 1, u, v, t, w, r)$$
that a cyclic shift of the elements has occurred, of order eight.
Thus $G\cong \Bbb Z_8$ because all cyclic groups of a given order are isomorphic.
The table is
$$\begin{array}{c|cccccccc} G & 1 & u & v & t & w & r & q & s \\ \hline 1 & 1 & u & v & t & w & r & q & s \\ u & u & v & t & w & r & q & s & 1 \\ v & v & t & w & r & q & s & 1 & u\\ t & t & w & r & q & s & 1 & u & v\\ w & w & r & q & s & 1 & u & v & t\\ r & \color{purple}r & \color{purple}q & \color{purple}s & \color{purple}1 & \color{purple}u & \color{purple}v & \color{purple}t & \color{purple}w\\ q & \color{orange}q & \color{orange}s & \color{orange}1 & \color{orange}u & \color{orange}v & \color{orange}t & \color{orange}w & \color{orange}r\\ s & s & 1 & u & v & t & w & r & q\\ \end{array}.$$
This was found by cyclically permuting the rows further and starting with the identity row for aesthetic reasons.