@@ -225,27 +225,27 @@ \section{Group actions}\label{sec:group_actions}
225225 We have listed the functions permuting \( a \) , \( b \) and \( ab \) in \cref {tab:ex:def:automorphism_group/v4 }.
226226
227227 \begin {table }[!ht]
228- \begin {center }
229- \begin {tabular }{l l l l}
228+ \begin {equation* }
229+ \begin {array }{l l l l}
230230 \toprule
231- Cycle in \( S_3 \) & \( a \) & \( b \) & \( ab \) \\
231+ \T { Cycle in} S_3 & a & b & ab \\
232232 \midrule
233- \( \ id \) & \( a \) & \( b \) & \( ab \) \\
234- \( \ cycle { 1, 2 } \) & \( b \) & \( a \) & \( ab \) \\
235- \( \ cycle { 1, 3 } \) & \( ab \) & \( b \) & \( a \) \\
233+ \id & a & b & ab \\
234+ \cycle { 1, 2 } & b & a & ab \\
235+ \cycle { 1, 3 } & ab & b & a \\
236236 \bottomrule
237- \end {tabular }
237+ \end {array }
238238 \quad
239- \begin {tabular }{l c c c}
239+ \begin {array }{l c c c}
240240 \toprule
241- Cycle in \( S_3 \) & \( a \) & \( b \) & \( ab \) \\
241+ \T { Cycle in} S_3 & a & b & ab \\
242242 \midrule
243- \( \ cycle { 2, 3 } \) & \( a \) & \( ab \) & \( b \) \\
244- \( \ cycle { 1, 2, 3 } \) & \( b \) & \( ab \) & \( a \) \\
245- \( \ cycle { 1, 3, 2 } \) & \( ab \) & \( a \) & \( b \) \\
243+ \cycle { 2, 3 } & a & ab & b \\
244+ \cycle { 1, 2, 3 } & b & ab & a \\
245+ \cycle { 1, 3, 2 } & ab & a & b \\
246246 \bottomrule
247- \end {tabular }
248- \end {center }
247+ \end {array }
248+ \end {equation* }
249249 \caption {Isomorphisms of the \hyperref [def:klein_four_group]{Klein four-group} \( V_4 \) }\label {tab:ex:def:automorphism_group/v4 }
250250 \end {table }
251251
@@ -1157,14 +1157,16 @@ \section{Group actions}\label{sec:group_actions}
11571157 \begin {equation* }
11581158 \begin {array }{l !{\quad } *{6}{l}}
11591159 \toprule
1160- \mdiagbox {\tau }{\sigma } & \id & \cycle { 1, 2 } & \cycle { 1, 3 } & \cycle { 2, 3 } & \cycle { 1, 2, 3 } & \cycle { 1, 3, 2 } \\
1160+ & \multicolumn {6}{c}{\sigma } \\
1161+ \cmidrule {2-7}
1162+ \multicolumn {1}{c}{\tau } & \id & \cycle { 1, 2 } & \cycle { 1, 3 } & \cycle { 2, 3 } & \cycle { 1, 2, 3 } & \cycle { 1, 3, 2 } \\
11611163 \midrule
1162- \id & \id & \cycle { 1, 2 } & \cycle { 1, 3 } & \cycle { 2, 3 } & \cycle { 1, 2, 3 } & \cycle { 1, 3, 2 } \\
1163- \cycle { 1, 2 } & \cycle { 1, 2 } & \id & \cycle { 1, 3, 2 } & \cycle { 1, 2, 3 } & \cycle { 2, 3 } & \cycle { 1, 3 } \\
1164- \cycle { 1, 3 } & \cycle { 1, 3 } & \cycle { 1, 2, 3 } & \id & \cycle { 1, 3, 2 } & \cycle { 1, 2 } & \cycle { 2, 3 } \\
1165- \cycle { 2, 3 } & \cycle { 2, 3 } & \cycle { 1, 3, 2 } & \cycle { 1, 2, 3 } & \id & \cycle { 1, 3 } & \cycle { 1, 2 } \\
1166- \cycle { 1, 2, 3 } & \cycle { 1, 2, 3 } & \cycle { 1, 3 } & \cycle { 2, 3 } & \cycle { 1, 2 } & \cycle { 1, 3, 2 } & \id \\
1167- \cycle { 1, 3, 2 } & \cycle { 1, 3, 2 } & \cycle { 2, 3 } & \cycle { 1, 2 } & \cycle { 1, 3 } & \id & \cycle { 1, 2, 3 } \\
1164+ \id & \id & \cycle { 1, 2 } & \cycle { 1, 3 } & \cycle { 2, 3 } & \cycle { 1, 2, 3 } & \cycle { 1, 3, 2 } \\
1165+ \cycle { 1, 2 } & \cycle { 1, 2 } & \id & \cycle { 1, 3, 2 } & \cycle { 1, 2, 3 } & \cycle { 2, 3 } & \cycle { 1, 3 } \\
1166+ \cycle { 1, 3 } & \cycle { 1, 3 } & \cycle { 1, 2, 3 } & \id & \cycle { 1, 3, 2 } & \cycle { 1, 2 } & \cycle { 2, 3 } \\
1167+ \cycle { 2, 3 } & \cycle { 2, 3 } & \cycle { 1, 3, 2 } & \cycle { 1, 2, 3 } & \id & \cycle { 1, 3 } & \cycle { 1, 2 } \\
1168+ \cycle { 1, 2, 3 } & \cycle { 1, 2, 3 } & \cycle { 1, 3 } & \cycle { 2, 3 } & \cycle { 1, 2 } & \cycle { 1, 3, 2 } & \id \\
1169+ \cycle { 1, 3, 2 } & \cycle { 1, 3, 2 } & \cycle { 2, 3 } & \cycle { 1, 2 } & \cycle { 1, 3 } & \id & \cycle { 1, 2, 3 } \\
11681170 \bottomrule
11691171 \end {array }
11701172 \end {equation* }
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