Definition of Generating set of a group

Question

I am trying to understand what group generators are. In the wikipedia article of the same title, the description given is, at least to me, extremely convoluted. It is said:

"In other words, if S is a subset of a group G then $\langle S \rangle$, the subgroup generated by S is the smallest subgroup of G containing every element of S , which is equal to the intersection over all subgroups containing the elements of S"

My first question is: What is a subset of a group? I know what a subgroup of a group is. But what about a subset?

If $\langle S \rangle$ contains every element of S, what is the distinction here?

Also when it says: "...which is equal to the intersection over all subgroups containing the elements of S" who is equal to the intersection?

Can someone explain to me what is being said here?

Answer 1

Group is set of elements (with operation). Subset of a group is just subset of this set.

The definition you should read in two parts:

1. $\langle S \rangle$ is subgroup of $G$ and contains every element of $S$
2. it is the smallest such subgroup

Then, there is a statement (in fact, equivalent definition): $\langle S\rangle$ is intersection of all subgroups containing all elements from $S$.

The difference between $S$ and $\langle S\rangle$ is that $S$ isn't necessarily a subgroup.

Answer 2

Have you learned linear algebra? If so, think about a vector subspace spanned by a set of vectors. It is roughly the same idea translated to a more abstract setting like groups.