TITLE: What is the difference between a Subgroup and a subset?
QUESTION [11 upvotes]: What is the difference between a Subgroup and a subset? I know hardly any Abstract algebra, just some things from youtube and wikipedia, but the notion of a subgroup being part of a larger group and a set being part of some group is indistinguishable to me. A nice simple answer would do fine. Thank you for your time

REPLY [7 votes]: You have laws that make a set $S$ become a group. You must have an operation $\star$ associated with this set which is closed under this operation (if $x,y\in S$ then $x \star y \in S$) and the operation must have these three properties :

If $a,b,c\in S$ then $a\star(b\star c)=(a\star b)\star c$
There exists a (unique) identity $e_S$ such that $a\star e_S=e_S\star a=a$ for every $a\in S$
For every $a\in S$, there exists a (unique) inverse $b$  such that $a\star b=b\star a=e_S$

If all these conditions are filled, then $S$ is called a group.
A subset of $S$ is just a subset of $S$ as a set.
But a subgroup of $S$ must have the three properties above.