- #1

mathmari

Gold Member

MHB

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I want to make the diagram for the dihedral group $D_6$:

Subroups of order $2$ : $\langle \tau \rangle$, $\langle \sigma\tau\rangle$, $\langle\sigma^2\tau\rangle$, $\langle\sigma^3\tau\rangle$, $\langle\sigma^4\tau\rangle$, $\langle\sigma^5\tau\rangle$, $\langle\sigma^3\rangle$

Subgroups of order $3$ : $\langle\sigma^2\rangle$, $\langle\sigma^4\rangle$

Subgroups of order $6$ : $\langle \sigma \rangle$, $\langle \sigma^5\rangle$, $\langle\sigma^2, \tau\rangle$, $\langle\sigma^2, \sigma\tau\rangle$

Are there more for each order? (Wondering)

The subgroups of order $4$ are those that are isomorphic to $\mathbb{Z}_4$ or to $\mathbb{Z}_2\times\mathbb{Z}_2$, right? (Wondering)

There are no elements of order $4$, so there are no subgroups of order $\mathbb{Z}_4$, right? (Wondering)

Are the subgroups that are isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$ the $\langle a,b\rangle$, for all $a, b$ the elemennt of order $2$

($a,b \in \{\tau, \sigma\tau, \sigma^2\tau , \sigma^3\tau , \sigma^4\tau, \sigma^5\tau , \sigma^3\}$ ) ? (Wondering)