Name of the theorem: If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$? Note: I am not asking for a proof of this theorem or any other theorem or help with a mathematical problem. This question is a reference request.
I use the following well-known and somewhat-easy-to-prove theorem in a paper but I don't want to waste space proving it.

Let $G$ be a group and $q$ be a prime power. If $q$ divides $|G|$, then $G$ has a subgroup of order $q$.

Essentially, I would just like to say something like "4 divides $|G|$. So $G$ has a subgroup of order $4$ by A's theorem." However, I haven't found any name for this theorem that I could use. An accepted answer to another question on MSE refers to this as "Sylow's First Theorem". However, MathWorld and Wikipedia say Sylow's theorems actually only state the existence of a subgroup of order $p^n$ where $n$ is maximal. This is also not Cauchy's theorem.
The proof of Sylow's theorem in Wikipedia and some other proofs use this theorem but I haven't found anything that refers to it by a name.
Does it have a name? It would also be okay to refer to another more powerful theorem if this theorem is directly implied. However, I wouldn't like referring for example to Sylow's theorem, because it still takes some steps to get from Sylow's theorem to this.
 A: From what I understand there isn't always universal agreement on how to name results. The theorem you refer to is indeed often given in a section on Sylow Theory, and maybe you can find it in Sylow's original paper where he does have a theorem $1.$ There is a discussion of the proof(s) here. Also, it seems that some books just talk about Sylow's theorem and not three separate theorems. Looking at the original paper, it looks like Sylow had $8$ theorems.
I checked a couple of textbooks:


*

*Gallian does call this Sylow's First Theorem and attributes the result to him.

*I couldn't quickly find it in Lang's Algebra book.

*Rotman's Advanced Modern Algebra calls the result a proposition but doesn't give the result a specific name.

*I couldn't quickly find the result in Hungerford's algebra book.


In summary, I don't think the result has a universally recognized name, but I doubt that many will be offended if you call it Sylow's theorem or Sylow's First Theorem or if you actually figure out what number it has in the original paper. If you need a reference to this, then I would just call it a theorem and then make sure to mention that it is due to Sylow.
