A doubt about Proposition $27$ in textbook Algebra by Saunders MacLane and Garrett Birkhoff

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I'm reading Proposition 27 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.

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If $N \trianglelefteq G$ and $S \subseteq G$, the join $N \vee S$ consists of all products $ns$ for $n \in N$ and $s \in S$. If both $N \cap S = \{1\}$ and $N \lor S=G$, then $G / N \cong S$.

Because the result is $G / N \cong S$, I think $S$ must be a subgroup, not just a subset of $G$. As such, I think it should be clearer to write $S \le G$, i.e. $S$ is a subgroup of $G$, rather than $S \subseteq G$.

Could you please verify if my observation is fine?

Update: I added the part that the authors define subgroup. I still feel that the use of $\subseteq$ for both subset and subgroup is confusing.

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Best Answer

In the remarks following Proposition 27, the authors state

In such cases, when $N \cap S = 1$ and $N \vee S = G,$ the subgroup $S$ is called a "complement" of the normal subgroup $N.$

This makes it clear that they intended $S$ to be a subgroup. As Arturo points out, the proposition is not valid without this assumption.

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