How many 2-Sylow subgroups does the group $D_n$ have?
We're trying to solve it by considering the case n is odd and n is even, but stucked in the case that n is odd.
Thank you very much!
Problem solved, thanks to Mark Bennett!
2026-03-26 12:45:19.1774529119
How many 2-Sylow subgroups does the group $D_n$ have?
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Solution:
|$D_n$| = 2n if n is an odd, the largest power of 2 that div |$D_n$| is 2, and we can find a 2-Sylow subgroup K={1, E}. By proving $N_{D_n}(K)=2$, we get $n_2$($D_n$)=2n/2=n.
if n is an even, we can construct a 2-Sylow subgroup K with size $2^m$(the largest power of 2 that div |$D_n$|). Then by proving K is a normal subgroup, we get it's the unique 2-Sylow subgroup and $n_2$($D_n$)=1.