The number of subgroups of \(D_{20}\) are-
(a) 20
(b) 40
(c) 48
(d) 50
Explanation: (c) is the correct answer as \(D_n\) has \(\tau(n)+\sigma(n)\) subgroups.
Note: If \(n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}\) then
\(\tau(n)=(k_1+1)(k_2+2)\cdots(k_r+r)\) and
\(\sigma(n)=\frac{p_1^{k_1+1}-1}{p_1-1}\frac{p_2^{k_2+1}-1}{p_2-1}\cdots \frac{p_r^{k_r+1}-1}{p_r-1}\)
Tags:
Groups