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Il gruppo diedrale e i suoi sottogruppi/main.tex b/Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.tex index 91e853c..4bae04f 100644 --- a/Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.tex +++ b/Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.tex @@ -106,7 +106,7 @@ per ogni $d \mid n$ esiste un unico sottogruppo di ordine $d$ dal momento che $\rotations$ è ciclico. Pertanto ogni tale sottogruppo assume la forma $\gen{r^{\frac{n}{d}}}$. Inoltre, dal momento che\footnote{ - Infatti ogni elemento di $D_n$, come visto prima, è della $r^k$ o $s r^k$. + Infatti ogni elemento di $D_n$, come visto prima, è della forma $r^k$ o $s r^k$. } $[D_n : \rotations] = 2$, $\rotations$ è un sottogruppo normale di $D_n$. Allora, poiché $\rotations$ è normale in $D_n$ e ogni sottogruppo $H \leq \rotations$ è caratteristico\footnote{ @@ -142,5 +142,59 @@ In conclusione, ogni sottogruppo di $D_n$ è della forma $\gen{r^d}$ o della forma - $\gen{r^d, sr^k}$. + $\gen{r^d, sr^k}$. Si mostra adesso che per $0 \leq k < d < n$ e $d \mid n$ la + classificazione è unica e completa. Chiaramente è completa, dal momento che + $d$ si può sempre ridurre in modulo $n$ e che per $k>d$ si può moltiplicare + $sr^k$ per riottenere una riflessione con esponente minore di $d$. \medskip + + + Si verifica adesso che è vi è un solo modo di esprimere un sottogruppo in queste + condizioni. Se $H_1 := \gen{r^{d_1}, s r^{k_1}} = \gen{r^{d_1}, s r^{k_1}} =: H_2$, allora + in particolare $H_1 \cap \rotations = \gen{r^{d_1}} = \gen{r^{d_2}} = H_2 \cap \rotations$, + da cui si deduce facilmente che $d_1 = d_2$. Sia ora $s r^{k_1} r^{t d_1} = + s r^{k_2} r^{t' d_2}$ con $t$, $t' \in \ZZ$. Allora deve valere che: + \[ r^{k_1 + t d_1} =r^{k_2 + t' d_2} \iff k_1 + t d_1 \equiv k_2 + t' d_2 \pod n, \] + e quindi, se $d := d_1 = d_2$, questo implica che: + \[ k_1 \equiv k_2 \pod d \implies k_1 = k_2 \impliedby 0 \leq k_1, k_2 < d. \] + Pertanto esistono esattamente\footnote{ + La scrittura $d(n)$ indica il numero di divisori di $n$, mentre $\sigma(n)$ indica la + somma dei divisori di $n$. Infatti per ogni divisore $d$ di $n$ si conta un + sottogruppo ciclico $\gen{r^d}$ e un sottogruppo della forma $\gen{r^d, sr^h}$ + con $0 \leq h < d$ (e quindi $d$ sottogruppi di questo tipo). + } $d(n) + \sigma(n)$ sottogruppi di $D_n$. \bigskip + + + + Si studiano adesso i sottogruppi normali di $D_n$. Come già visto, i sottogruppi di + $\rotations$ sono tutti normali in $D_n$. Si studiano dunque soltanto i gruppi + della forma $\gen{r^d, s r^h}$. Si ricorda che il sottogruppo + $H = \gen{r^d, s r^h}$ è normale se e solo se $N_G(H) = G$, ossia se il suo + normalizzatore è tutto $G$. In particolare questo è vero se i generatori di $G$ + appartengono a $N_G(H) = G$ e quindi se $r H r\inv = H$ e se $s H s\inv = H$. + In particolare\footnote{ + Si è utilizzata la relazione $g \gen{g_1, \ldots, g_i} g\inv = \gen{g g_1 g\inv, \ldots, g g_i g\inv}$. + } si deve studiare quando valgono le seguenti identità: + \[ + \gen{r^d, s r^h} = \underbrace{\gen{r^d, s r^{h-2}}}_{r H r\inv}, + \qquad \gen{r^d, s r^h} = \underbrace{\gen{r^{-d}, r^h s\inv}}_{s H s\inv} = + \gen{r^d, s r^{-h}}. + \] + La prima identità è vera se e solo se $h \equiv h-2 \pod d$, e quindi se e solo + se $d \mid 2$. Pertanto $d$ può valere solo $1$ o $2$: se $d = 1$, $H$ è + esattamente $\gen{r, s} = D_n$; se invece $d = 2$, $H$ può essere soltanto + $\gen{r^2, s}$ o $\gen{r^2, s r}$. Ciononostante, nel caso $d = 2$, dal momento + che $d \mid n$, $n$ deve essere pari (e quindi se $n$ è dispari, $D_n$ non ammette + sottogruppi normali non banali, ed è in particolare semplice). \medskip + + + Si consideri dunque $n$ pari. È sufficiente controllare che per $d = 2$ valga + anche la seconda identità, ossia che valga $h \equiv -h \pod 2$, sempre verificata. + Si conclude dunque con la seguente classificazione: + + \begin{itemize} + \item se $n$ è dispari, $D_n$ ammette come sottogruppi normali soltanto + $D_n$, $\{ e \}$ e i sottogruppi di $\rotations$, + \item se $n$ è pari, $D_n$ ammette come sottogruppi normali tutti quelli + del caso dispari insieme a $\gen{r^2, s}$ e $\gen{r^2, s r}$. + \end{itemize} \end{document} \ No newline at end of file