The argument on page 105, starting from the paragraph “We define a homomorphism…” may be harder to follow than we thought, and so let us give more details here.<\/p>\n
Given $H\\leq \\mbox{Sym}(\\Delta)$ and $B=\\mbox{Func}(\\Delta, G)$, define a map $\\tau:H\\to \\mbox{Aut}(B)$ as follows: for $f\\in B$, $f(h\\tau)$ is the map
\n\\[
\nf(h\\tau): \\delta\\mapsto (\\delta h^{-1})f,\\quad\\mbox{or equivalently}\\quad \\delta(f(h\\tau))= (\\delta h^{-1})f.
\n\\]<\/p>\n
To prove that $f(h\\tau)$ lies in $\\mbox{Aut}(B)$ is routine but it involves proving that $h\\tau$ is a bijection, and that $(f_1f_2)(h\\tau)=(f_1(h\\tau))(f_2(h\\tau))$, noting that multiplication in $B$ is pointwise.<\/p>\n
To prove that $\\tau$ is a homomorphism, take $h, h’\\in H$. Then $(h\\tau)\\circ (h’\\tau)$ maps an arbitrary $f\\in B$ to $f_1(h’\\tau)$, where $f_1=f(h\\tau)$. (Note that $\\circ$ denotes composition of automorphisms in $\\mbox{Aut}(B)$.) Using the display above we have
\n\\begin{align*}
\n\\delta(f((h\\tau)\\circ (h’\\tau)))&= \\delta(f_1((h’\\tau)))\\\\
\n&=(\\delta (h’)^{-1})f_1 \\\\
\n%=(\\delta (h’)^{-1})(f(h\\tau)) \\\\
\n&= (\\delta (h’)^{-1}h^{-1})f \\\\
\n&= (\\delta (hh’)^{-1})f\\\\
\n&=\\delta(f((hh’)\\tau).
\n\\end{align*}
\nSince this holds for all $\\delta\\in\\Delta$ we have $f((h\\tau)\\circ (h’\\tau)) = f((hh’)\\tau)$, and since this holds for all $f\\in B$ we have $(h\\tau)\\circ (h’\\tau) = (hh’)\\tau$. Hence $\\tau$ is a homomorphism.<\/p>\n","protected":false},"excerpt":{"rendered":"
The argument on page 105, starting from the paragraph “We define a homomorphism…” may be harder to follow than we thought, and so let us give more details here. Given $H\\leq \\mbox{Sym}(\\Delta)$ and $B=\\mbox{Func}(\\Delta, G)$, define a map $\\tau:H\\to \\mbox{Aut}(B)$ as follows: for $f\\in B$, $f(h\\tau)$ is the map \\[ f(h\\tau): \\delta\\mapsto (\\delta h^{-1})f,\\quad\\mbox{or equivalently}\\quad … Continue reading Page 105<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":71,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1164"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=1164"}],"version-history":[{"count":6,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1164\/revisions"}],"predecessor-version":[{"id":1185,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1164\/revisions\/1185"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/71"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=1164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}