Statement: Let $G$ be a finite group. Then $ I_2(G)=[G:G^2]-1$.
where; $I_2(G)$ is the number of subgroups of index two in $G$ , $[G,G^2]$ represents the index of $G^2$ in $G$ and $G^2= \langle \{x^2: x\in G\} \rangle$.
My work: First I proved that $G^2$ is the subgroup of $G$ generated by squares of elements in $G$ and also $G^2$ is normal in $G$. I found that $\frac{G}{G^2}$ is abelian.
Then, I took for instance, $G=D_n=\langle R,S:$ $R$ is a rotation with $o(R)=n$ and $S$ is reflection$\rangle $ so that $G^2= \langle R^2 \rangle$. Which tells that the above statement is true for $D_n$.
My Question: I thought a lot for the proof of this statement but I am not getting any useful tool\concept to prove this.
Please provide a proof of this statement.
from Hot Weekly Questions - Mathematics Stack Exchange
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