Group and subgroup I just learned that a group is a set together with an operation such as $<G,•>$. 
And a subgroup is a subset H of G, which forms a group under "•". 
I got confused by the definitions because I think subgroup should be a group and it should be $<H,•>$. Is that right?
 A: You're absolutely correct.
(But, be aware that groups and subgroups are often referred to just as, for example, $G$, without specifying an operation, when the writer assumes that you will know what operation is being talked about.)
(Specifically, if we talk about "a subgroup $H$ of $<G, \cdot>$", the operation could only be $\cdot$, so we don't need to say "a subgroup $<H, \cdot>$ ..." even though that's really what we mean.)
A: Correct, a group is a set with 3 properties


*

*Has an identity element

*Is closed under the operation • (whatever • may be)



*Is closed under inverses



An important point to note is that H is not just any group but one that borrows the identity element and operation from G. It's properties:


*

*Has the same identity element as G

*Is closed under the same operation as the one in G

*Is closed under inverses

A: You are saying "a subgroup $H$ should be a group...". I will pause here, and ask under what operation? We expect that it should be group under same operation as in $G$. 
The same is said in first two lines: $H$ is a subset, such that it becomes group under same $\bullet$ as in $G$. 
