What does this definition mean ?

[Maths Lover]

hello , I don't understand the meaning of the next definition , so , I hope that you can make it easy to understand it for me

definition
if G is an arbitrary group and ∅≠S⊆G , then ,the symbol (s) will represent the set
(S) = ∩ { H∖S ⊆ H : H is a subgroup of G }

can you give me some examples about this definition ?

 

[Stephen Tashi]

definition
if G is an arbitrary group and ∅≠S⊆G , then ,the symbol (s) will represent the set
(S) = ∩ { H∖S ⊆ H : H is a subgroup of G }

Perhaps the definition should read: (S) = ∩ { H | S ⊆ H : H is a subgroup of G }
where "H |" means "the H such that...".

(The notation "H\G" is used by some books to denote the set of right cosets of a subgroup H. I don't think that is what the definition uses.)

The usual definition of (S) would be called "the subgroup generated by S" and it would be
$\cap_i H_i$ taken over all subgroups $H_i$ of $G$ that contain $S$ as a subset.

What examples of finite groups have you studied? Perhaps we can use one of them as an example.