With two subgroups of a group $G$, does one being a subset of the other imply being a subgroup of it?
Let $(G,\ast,e)$ be a group, and $(H,\ast,e)$ and $(K,\ast,e)$ be subgroups of $G$. If $K\subseteq H$, is $K \leq H$ ?
With the subgroup test, it seems trivial, but I just want to make sure I'm not making some stupid mistake, as I'm using this for another proof that's causing me to question everything.
$\endgroup$ 22 Answers
$\begingroup$Yes, that is true. It follows immediately from one step subgroup test, that is
For K subset of H,
K is a subgroup of H if and only if:
- K is not empty;
- For every two elements $a,b\in K$ also $a^{-1}b\in K$
Can you check that these two conditions hold?
$\endgroup$ 2 $\begingroup$Assuming a fixed operation $*$ which is the same for all groups in the context, the statement $K$ is a subgroup of $H$ just means that (i) $K$ is a subset of $H$ AND (ii) $K$ is a group.
This applies in this case since $G, H, K$ are all using the same operation. Since $K$ is a subgroup of $G$, it's a group, and since it's a subset of $H$, by the above fact, it's a subgroup of $H$.
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