I have to find maximal normal $2$-subgroup of the dihedral group $D_{20}$, of order $40.$ According to me as Sylow $2$-subgroups are isomorphic to $D_4$, are $5$ in numbers so not normal. Center of group is $C_2$ i.e. cyclic group of order $2$ is normal. There are also subgroups of order $4$, $C_4$ and $C_2\times C_2$ but i do not know that these are normal or not. Please suggest me. Thanks.

  • $\begingroup$ Firstly, the maximum normal 2-subgroup has order 4, not 2. Secondly, the group $D_{20}$ has subgroups of order $2$ that are not normal. And thirdly, the maximal normal $2$-subgroup of any finite group is unique, so there can only be one answer, not several. $\endgroup$ – Derek Holt Jan 25 at 17:16
  • $\begingroup$ @DerekHolt thanks ......can you please tell me which subgroup is maximal normal $2$-subgroup. $\endgroup$ – neelkanth Jan 25 at 17:18
  • $\begingroup$ Here $D_{20}$, is of order $40.$ $\endgroup$ – neelkanth Jan 25 at 17:19
  • $\begingroup$ @DerekHolt Can you tell me which subgroup is that one.... $\endgroup$ – neelkanth Jan 25 at 17:40
  • $\begingroup$ I think $C_4$ is the required one ? $\endgroup$ – neelkanth Jan 25 at 17:55

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