Suppose $G$ is a group under addition. Must it be the case that the subgroup $H$ inherits the same product operation as $G$? For example, if $G = \mathbb{R}$ and $H$ is a proper subset of $\mathbb{R}$ with a group structure under multiplication, is $H$ still a subgroup of $G$?
This topic doesn't seem to be covered in any of the textbooks I am working in, perhaps because it is considered obvious. In this particular case, the identity and inverse elements would differ, but in theory there could be a case where the operation differs but the identity and inverses are preserved. (Or is there?)
I suppose I am curious on whether the product operation is preserved by definition or by necessity (i.e., we can't find an example where it isn't) or whether this isn't even a requirement to begin with.
Best Answer
A subgroup must inherit the same group operation as the supergroup, as a matter of definition. For example, one could easily define modulo $4$ arithmetic on the subset $\{0, 1, 2, 3\}$ of $\Bbb{R}$ (under addition), but it's not a subgroup, because $3$ and $1$ do not add to $0$ in $(\Bbb{R}, +)$, but they do in modulo $4$ arithmetic.
One could do something similar with $\{1, 2, 3, 4\}$, or $\{8, -1, \pi, 2.1\}$. Note that the group structure of $(\Bbb{R}, +)$ has nothing to do with it. Really, the only structure necessary to form this relaxed type of "subgroup" is for the set to have cardinality less than or equal to $\Bbb{R}$. That is, the formation of these "subgroups" is less about how they interact via $+$ and more about the number of elements in $\Bbb{R}$.