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I am interested the following problem: Let $p$ be a prime. Determine the subgroups of $\mathbb{Z}/(p-1)\mathbb{Z}\times \mathbb{Z}/(p-1)\mathbb{Z}$ of index 2 in terms of their generators.

I'm trying to use a result that says subgroups of $\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ are in bijection with triples $(a,b,i)$ with $a,b|n$ and $0\leq i < \gcd(a,b)$. (See Andrew Sutherland's paper Computing Images of Galois Representations Attached to Elliptic Curves, Lemma 3.4 for details on the bijection). This forces $a$ to be even and $b=a/2$ but actually getting my hands on the generators is tough using this method.

If anybody had any way of using this method or another method to determine the subgroups of index 2, I would be grateful.

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If $H$ is a subgroup of index $2$ in $G$, then $H$ must contain $G^2=[G,G]G^2$, the first term of the lower exponent-$2$ central series.

Now, suppose $G=\langle a\rangle \times \langle b\rangle$ with $a$ and $b$ having even order. (As in your case, when $p$ is odd.) In this case, $G^2=\langle a^2,b^2\rangle$ and $G/G^2$ is isomorphic to the Klein group. By the correspondence theorem, there are three choices for $H$: $\langle a^2,b^2, a \rangle=\langle a,b^2\rangle$, $\langle a^2,b^2, b \rangle=\langle a^2,b\rangle$ and $\langle a^2,b^2, ab \rangle=\langle ab,b^2 \rangle$.

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