# What does the notation $\langle x^n:\; x\in G\rangle$ mean?

If $G$ is a group, what does $$G^n:=\langle x^n:\; x\in G\rangle$$ for a fixed $n\in\mathbb{N}$ mean? This should be a subgroup of $G$ but I din't know the definition of $\langle x^n:\; x\in G\rangle$, The notation is new for me.

The background of this question is, I'm interested in this A group is divisible if and only if it has no maximal subgroup ? answer .

Regards

The question above is answered in the comments above, thanks! I have an additional question:

Here https://en.wikipedia.org/wiki/Generating_set_of_a_group wikipedia says that the elements of $G^n$ can be expressed as a finite product of elements in $S$ and their inverses I find a description here Subgroup generated by a set ). WIth this characterisation, how to write $G^n$? I thought $\{x^nx^{-n}:x\in G\}$ first, but it doesn't make sense.

Edit 2:The additional question is clear now!!

• If you have a subset of a group $S \subseteq G$, then $\langle S \rangle$ denotes the subgroup of $G$ generated by $S$. – Crostul Oct 7 '15 at 7:08
• It looks like $n$ is a fixed integer, and this means the group generated by the $n$ths powers of elements of $G$. The angled brackets, $\langle\qquad\rangle$, mean the structure (here obviously a subgroup) generated by whatever set is inside. The colon explains the use of variables much like in set construction notation. – Jyrki Lahtonen Oct 7 '15 at 7:09
• with $S=\{x^n: x\in G\}$? – algebra Oct 7 '15 at 7:09
• @algebra Exactly. – Ben Sheller Oct 7 '15 at 7:10
• okay thanks. Here en.wikipedia.org/wiki/Generating_set_of_a_group wikipedia says that the elements of $G^n$ can be expressed as a finite product of elements in $S$ and their inverses I find a description here math.stackexchange.com/questions/145442/… ). WIth this characterisation, how to write $G^n$? I thought $\{x^nx^{-n}:x\in G\}$ first, but it doesn't make sense. – algebra Oct 7 '15 at 7:22