# The set $\langle S\rangle$ is called the subgroup generated by $S$. Please help explain this.

Complete beginner to abstract algebra and the concepts are very foreign (do all newcomers to this subject feel this way?)

In any case Contemporary Abstract Algebra by Gallian tries to explain the notion of a "subgroup generated by $S$" in one paragraph. I like the book but he lists examples without going into sufficient detail for me to pick up what he is trying to say (note...I've reread this several times over a couple of days trying to pick up the concept). I don't think I need a lot of help just a sentence or two to explain to give me that "aha, " moment. The related/similar questions seem to involve proofs. I don't want that...I just want the concept explained a bit more.

In any case the book states:

For any element $a$ of a group $G$ it is useful to think of $\langle a\rangle$ as the smallest subgroup of $G$ containing $a$. This notion can be extended to any collection $S$ of elements from a group $G$ by defining $\langle S\rangle$ as the subgroup of $G$ with the property that $\langle S\rangle$ contains $S$ and if $H$ is any subgroup of $G$ containing $S$, then $H$ also contains $\langle S\rangle$. Thus, $\langle S\rangle$ is the smallest subgroup of $G$ that contains $S$. The set $\langle S\rangle$ is called the subgroup generated by $S$."

The author then goes on to list examples without any explanation:

• In $Z_{20} \langle 8,14\rangle = \{0, 2, 4, ..., 18\} = \langle 2\rangle$.
• In $Z\langle 8, 13\rangle = \mathbb{Z}$.
• In $D_{4} \langle H, V\rangle = \{H, H^2, V, HV \} = \{R_{0}, R_{180}, H, V \}$.

Thank you in advance...

-IdleMathGuy

• What exactly is unclear to you in that quotation? – Dalamar Sep 6 '17 at 13:53
• An alternative way of thinking about $\langle S \rangle$ is "all elements of $G$ that you can make out of the elements in $S$ by applying the group operation repeatedly". This, I think, explains why it is called the subgroup generated by $S$, as you can generate all its elements starting from the elements of $S$. – Magdiragdag Sep 6 '17 at 13:53
• Probably you know a similar concept from linear spaces: generating (spanning) subset of a linear space. Moreover in a linear space you have minimal generating subspaces (called bases) that are characterized by the fact that they have the same number of element (called dimension of the space) for finite-dimensional spaces. For group, minimal generating subsets do not have in general the same number of elements. – trying Sep 6 '17 at 14:01
• @Magdiragdag This was EXACTLY what I needed. Thank you both. Question...for the examples above do I have to use all of elements in the group to create the next group or can I just use one repeatedly? (or maybe not applicable..if I use <8, 14> in the first example above I could use $8^n$ or $14^n$ or 8 times 14, etc.) – Idle Math Guy Sep 6 '17 at 14:16